math.seu.edu.cn · abstract. main references: [1] choquer-bruhat, yvonne; dewitt-morette, c.;...

Analysis on manifolds

Yi Li

SCHOOL OF MATHEMATICS AND SHING-TUNG YAU CENTER, SOUTHEASTUNIVERSITY, NANJING, CHINA

E-mail address: [email protected]; [email protected]; [email protected]

ABSTRACT. Main references:

[1] Choquer-Bruhat, Yvonne; DeWitt-Morette, C.; Dullard-Bleick, M. Analy-sis, Manifolds and Physics, Part I: Basics, Revised Edition, Elsevier, 2010.

[2] Chandrasekhar, S. The Mathematical Theory of Black Holes, OUP Oxford,1998.

[3] Cheeger, Jeff; Ebin, David, G. Comparison Theorems in Riemannian Geome-try, American Mathematical Society, 2008.

[4] Yau, Shing-Tung; Schoen, Richard. Lectures on Differential Geometry (inChinese), Higher Education Press, 2004.

Contents

Chapter 1. Basic analysis 51.1. Set theory 51.2. Algebraic structures 71.3. Topology 111.4. Measures and integrations 241.5. Linear functional analysis 401.6. Differentiable calculus on Banach spaces 451.7. Calculus of variations 521.8. Implicit function theorem and inverse function theorem 541.9. Differentiable equations 591.10. Problems and references 61

Chapter 2. Differentiable manifolds 652.1. Differentiable manifolds 652.2. Vector fields and tensor fields 692.3. Lie groups 872.4. Exterior differential forms 1082.5. Integration 1192.6. Exercises and problems 130

Chapter 3. Riemannian manifolds 1313.1. Pseudo-Riemannian structures 1313.2. Linear connections 1423.3. Variations in Riemannian geometry 1593.4. Exponential maps and normal coordinates 1713.5. Second fundamental forms of geodesic spheres 1783.6. Comparison theorems 1843.7. Manifolds with nonnegative curvature 1953.8. Space of metric measure spaces 2003.9. Ricci flow 2003.10. Exercises and problems 200

Chapter 4. Kahler manifolds 2014.1. Complex manifolds 2014.2. Kahler manifolds 2014.3. Calabi-Yau manifolds 2014.4. Compact complex surfaces 2014.5. Kahler-Ricci flow 2014.6. Exercises and problems 201

3

CHAPTER 1

Basic analysis

1.1. Set theory

Given a set X. Let A be a subset of X. The characteristic function of A is

χA(x) :=

1, x ∈ A,0, x /∈ A.

A partition of X is a family Xii∈I of subsets such that

Xi = ∅, Xi ∩ Xj = ∅ (i = j),∪i∈I

Xi = X.

Here I is the indexed set which may be uncountable.

1.1.1. Categories. A category C is a class consisting of

Ob(C) = objects such like X, Y, · · · , ,Mor(C) = sets HomC(X, Y) : X, Y ∈ Ob(C) ,

where elements of HomC(X, Y) are called morphisms, together with composi-tions:

HomC(X, Y)×HomC(Y, Z) −→ HomC(X, Z), ( f , g) 7−→ g f ,

satisfying the following conditions:(i) h (g f ) = (h g) f ,

(ii) for each X ∈ Ob(C) there is a unique element 1X ∈ HomC(X, X) suchthat

f 1X = f , 1X g = g,for any f ∈ HomC(X, Y) and any g ∈ HomC(Y, X).

The simplest example is Set consisting of sets and set mappings.

A covariant functor C : A→ B consists of mappings

C : Ob(A) −→ Ob(B), X 7−→ C(X)

and

C : Mor(A) −→ Mor(B), f ∈ HomA(X, Y) 7−→ C( f ) ∈ HomB(C(X), C(Y)),

such thatC(g f ) = C(g) C( f ), C(1X) = 1C(X).

A contracovariant functor C : A→ B consists of mappings

C : Ob(A) −→ Ob(B), X 7−→ C(X)

and

C : Mor(A) −→ Mor(B), f ∈ HomA(X, Y) 7−→ C( f ) ∈ HomB(C(Y), C(X)),

5

6 1. BASIC ANALYSIS

such thatC(g f ) = C( f ) C(g), C(1X) = 1C(X).

1.1.2. Relations. A relation between sets X and Y is a subset R of X×Y. Write

(x, y) ∈ R ⇐⇒ xRy.

A relation R ⊂ X× X is an equivalence relation in X if

(i) (reflexive) ∀ x ∈ X =⇒ (x, x) ∈ R;(ii) (symmetric) ∀ x, y ∈ X and (x, y) ∈ R =⇒ (y, x) ∈ R;

(iii) (transitive) ∀ x, y, z ∈ X and (x, y) ∈ R, (y, z) ∈ R =⇒ (x, z) ∈ R.

If R is an equivalence relation, we write x ∼ y instead of xRy.

(1) The equivalence class of x ∈ X is

[x] := y ∈ X : y ∼ x.

Then X is the disjoint union of all [x] and denote

X/ ∼:= [x] : x ∈ X.

(2) Define x ∈ y in R by x− y = 2πn for some n ∈ Z. Then R/ ∼∼= S1 theunit circle on R2.

1.1.3. Orderings. A relation R ⊂ X× X is a partial ordering in X if

(i) (reflexive) ∀ x ∈ X =⇒ (x, x) ∈ R;(ii) (anti-symmetric) (x, y) ∈ R and (y, x) ∈ R =⇒ x = y;

(iii) (transitive) (x, y) ∈ R and (y, z) ∈ R =⇒ (x, z) ∈ R.

If R is a partial ordering, we write x ≤ y instead of xRy.

(1) A partially ordered set is a pair (P,≤), where ≤ is a partial ordering onP. Let a, b, c ∈ P.

(1.1) c is an upper bound for a and b if a ≤ c and b ≤ c.(1.2) c is the least upper bound or supremum of a and b, if c is an upper

bound for a and b, and satisfies c ≤ d for any upper bound d for aand b. Write

c = sup(a, b) = a ∨ b.

(1.3) Similarly, one can define a lower bound for a and b and the greatestlower bound or infimum c of a and b. Write

c = inf(a, b) = a ∧ b.

(2) A partially ordered set (P,≤) is directed if any pair of elements of P hasan upper bound.

(3) A partially ordered set (P,≤) is a lattice if for any a, b ∈ P, both sup(a, b)and inf(a, b) exist.

(4) Let (P,≤) is a partially ordered set, and m, p ∈ P. We say m is maximalif m ≤ p⇒ m = p.

(5) A partially ordered set (P,≤) is linearly ordered or totally ordered if forany a, b ∈ P, either a ≤ b or b ≤ a.

1.2. ALGEBRAIC STRUCTURES 7

Lemma 1.1.1. (Zorn’s Lemma) Let (P,≤) be a nonempty partially ordered set such thateach linearly ordered subset of P has an upper bound in P. Then for every x ∈ P there is amaximal element y ∈ P such that x ≤ y.

Exercise 1.1.2. (1) P = Z+ = N \ 0, and m ≤ n⇔m|n. Then (P,≤) is a partiallyordered set, and

sup(m, n) = [m, n], inf(m, n) = (m, n).(2) P = R and x ≤ y in the usual sense. Then (P,≤) is a partially ordered set,

andsup(x, y) = maxx, y, inf(x, y) = minx, y.

(3) P = 2U , where U is a set, and A ≤ B⇔ A ⊆ B. Then (P,≤) is a partiallyordered set and

sup(A, B) = A ∪ B, inf(A, B) = A ∩ B.(4) Given a nonempty set X, define P the set of all real functions on X, and

f ≤ g⇔ f (x) ≤ g(x) for all x ∈ X. Then (P,≤) is a partially ordered set.

PROOF. Obviously.

1.2. Algebraic structures

Let A be a subset of a set X. An internal operation on X is a mapping X×X →X, while an external operation on X by A is a mapping A× X → X.

1.2.1. Groups. A groups is a set X together with an internal operations (calledmultiplication)

X× X −→ X, (x, y) 7−→ xy,such that

(i) (associative) ∀ x, y, z ∈ X =⇒ (xy)z = x(yz);(ii) (identity) ∃ (hence ∃!) e ∈ X such that xe = ex = x for all x ∈ X;

(iii) (inverse) ∀ x ∈ X ∃! x−1 ∈ X such that xx−1 = e = x−1x.The group X is Abelian if xy = yx for all x, y ∈ X.

(1) Define a category Group as follows: objects are groups, and morphismsare group homomorphisms. That is, f ∈ HomGroup(G1, G2) if and onlyif f (x1y1) = f (x1) f (x2) for any x1, y1 ∈ G1.

(2) Let X be a group.(2.1) The center of X is the set x ∈ X : xy = yx, ∀ y ∈ X. It clearly

contains e.(2.2) A subset A ⊆ X is said to be a subgroup of X if xy−1 ∈ A whenever

x, y ∈ A. In this case we write A < X.(2.3) Suppose A < X and choose x ∈ X. Define the left coset and right

coset by

xA := xa : a ∈ A, Ax := ax : a ∈ A,respectively.

8 1. BASIC ANALYSIS

(2.4) A < X is said to be normal if xax−1 ∈ A for all a ∈ A and all x ∈ X.In this case we have Ax = xA and write A X.

(2.5) For A X define the quotient group by

X/A := xA : x ∈ Awith multiplication (xA)(yA) := (xy)A.

Exercise 1.2.1. (1) (Max(n, R),+) is an Abelian group, where Max(n, R) is the setof all n× n real matrices.

(2) (GL(n, R), ·) is a non-Abelian group, where GL(n, R) is the subset ofMax(n, R) with nonzero determinants.

(3) T := R/Z ∼= S1.(4) Show that Group is a category.

PROOF. (1) Obviously.(2) GL(n, R) is not Abelian under the matrix multiplication:[

1 23 4

] [5 67 8

]=

[19 2243 50

]=[

23 3431 46

]=

[5 67 8

] [1 23 4

](3) Consider the map

φ : T −→ S1, [r] := r + Z 7−→ e2π√−1r.

So φ is well-defined that is independent of the choice of r ∈ [r]. Conversely, wedefine ψ(e2π

√−1r := [r]. Then φ is bijective. Moreover

φ([r] + [s]) = φ([r + s]) = e2π√−1(r+s) = e2π

√−1re2π

√−1s = φ([r])φ([s])

so that φ is a group homomorphism. Hence φ is a group isomorphism.(4) Obviously.

1.2.2. Rings. A ring is a set X together with multiplication (x, y) 7→ xy, andaddition (x, y) 7→ x + y, such that

(i) (X,+) is Abelian (so has the identity 0);(ii) (associative and distributive):

(xy)z = x(yz), x(y + z) = xy + xz, (y + z)x = yx + zx.

A ring with an identity e (i.e., ex = xe = x for all x ∈ X) is called a ring withidentity.

(1) A ring is Abelian under multiplication is said to be a commutative ring.(2) Let X be a ring with identity. x ∈ X is said to be invertible if ∃ y ∈ X

such that xy = e = yx (show that such a y is unique!).(3) A ring with identity is called a field if ∀ 0 = X is invertible.(4) Usually we take K = R or C.

Let X be a ring.(1) A left ideal I of X is a subring of X such that ∀ x ∈ X and ∀ i ∈ I ⇒

xi ∈ I.(2) Similarly one can define a right ideal of X. An ideal of X is both left and

right.

1.2. ALGEBRAIC STRUCTURES 9

(3) Let I be an ideal of X. Define an equivalence relation by

x ∼ y ⇐⇒ x− y ∈ I.

Then the quotient

X/I :=[x] := x + i|i ∈ I

∣∣x ∈ X

is a ring, called a quotient ring of X:

[x][y] := [xy], [x] + [y] := [x + y].

1.2.3. Modules. A module X over R (ring) is an Abelian group X togetherwith an external operation (scalar multiplication):

R× X −→ X, (α, x) 7−→ αx,

such that(i) α(x + y) = αx + αy, ∀ x, y ∈ X and ∀ α ∈ R,

(ii) (α + β)x = αx + βx, ∀ x ∈ X and ∀ α, β ∈ R,(iii) (αβ)x = α(βx), ∀ x ∈ X and ∀ α, β ∈ R,(iv) ex = x, ∀ x ∈ R (if R is a ring with identity).

1.2.4. Algebras. An algebra A is a module over a ring R with identity, to-gether with an internal associative operation (called multiplication) such that

(i) A is a ring,(ii) α(xy) = (αx)y = x(αy), ∀ x, y ∈ A and ∀ α ∈ R.

1.2.5. Linear spaces. A linear space or vector space is a module X (over ringR) for which the ring of operators is a field (namely, R is a field). Usually andalways, we take R = K. Elements of X are called vectors. A subset L of X is calleda vector subspace if L is a module over the field K.

(1) Let X be a vector space, and L, M are vector subspaces. We say X is thedirect sum of L and M,

X = L⊕M,

if ∀ z ∈ X ∃! x ∈ L and y ∈ M such that z = x + y.(2) Let X be a vector space.

(2.1) A ⊂ X is called linearly independent if ∀ xi1≤i≤n ⊂ A with∑1≤i≤n λixi = 0, then λi = 0 for each 1 ≤ i ≤ n.

(2.2) A Hamel basis of X is a maximal linearly independent subset of X(the existence follows from Lemma 1.1.1).

(2.3) Two Hamel bases have the same cardinality, so that we can definethe dimension of X. That is, dim(X) := #Hamel basis.

(2.4) The codimension of a vector subspace L ⊂ X is codim(L) := dim(X \L).

(2.5) A subset A of X is convex if

x, y ∈ A, 0 ≤ λ ≤ 1 =⇒ λx + (1− λ)y ∈ A.

(2.6) An affine subspace of affine hyperplane of X is a set

x ∈ X : x = y + x0, y ∈ L,where x0 is a given vector of X and L is a vector subspace of X.

Let X, Y be vector spaces over K.

10 1. BASIC ANALYSIS

(1) A mapping f : X → Y is linear if

f (αx + βy) = α f (x) + β f (y)

for all x, y ∈ X and all α, β ∈ K. Define

Ker( f ) := x ∈ X : f (x) = 0.

Theorem 1.2.2. A linear mapping f is injective if and only if ker( f ) = 0.

Theorem 1.2.3. The inverse of a bijective linear mapping is also linear.

(2) Define

(1.2.1) L(X, Y) := linear mappings from X to Y.(3) Set

(1.2.2) X∗ := L(X, K)

the algebraic dual of X. Elements of X∗ are called linear forms or linearfunctionals.

(4) A sesquilinear mapping is a mapping

X× X −→ K, (x, y) 7−→ (x|y),satisfying

(x|y) = (y|x), (αx + βy|z) = α(x|z) + β(y|z).(5) A sesquilinear mapping is nondegenerate if

f : X −→ X∗, x 7−→ (x|·)is bijective. If dim(X) is finite, then

(1.2.3) nondegenerate ⇐⇒((y|x) = 0, ∀ x ∈ X ⇒ y = 0

).

(6) A sesquilinear mapping is positive if (x|x) ≥ 0, ∀ x ∈ X. It is strictlypositive if it is positive and ((x|x) = 0⇒ x = 0).

(7) A pre-Hilbert space is a pair (X, (·|·)), where X is a vector space and (·|·)is a strictly positive sesquilinear mapping.

Exercise 1.2.4. (1) Show that L(X, Y) is a vector space over K.(2) On X = R2 define

(x|y) := x1y1, x = (x1, x2), y = (y1, y2) ∈ R2.

Show that (·|·) is a degenerate sesquilinear mapping.(3) On X = C(U) (the set of all continuous functions on U), where U is a closed

and bounded interval of R, define

(y|x) :=∫

Uy(t)x(t)dt.

1.3. TOPOLOGY 11

Show that dim(X) = ∞, (·|·) is a degenerate sesquilinear mapping, and (1.2.3) isnot true in this case.

PROOF. (1) Obviously.(2) It suffices to check that (1.2.3) is not valid. Assume (x|y) = 0 for all x ∈ R2.

Then y1 = 0 and y = (0, y2). So we can not conclude that y = 0. Therefore (·|·) isdegenerate.

(3) If (y|x) = 0 for all x ∈ X, then taking x = y yields y = 0. It is clear that themapping

f : X −→ X∗, x 7−→ (x|·)is injective, however it is not surjective. Indeed, we can not find any x ∈ X suchthat f (x) = δ, where δ(y) := y(t0) is the delta function at some given point t0 ∈ D.Otherwise

y(t0) = δ(y) = (x|y) =∫

Dx(t)y(t)dt =⇒ x(t0) =

∫D|x(t)|2dt ≥ 0.

But x(t0) need not to be nonnegative. Hence (·|·) is degenerate.Since tn ∈ X, it follows that dim X = +∞.

1.3. Topology

1.3.1. Topology. A system T of subsets of a set X defines a topology on X if

(i) ∅, X ∈ T ,(ii) ∀ Ui ∈ T , i ∈ I =⇒ ∪i∈IUi ∈ T ,

(iii) ∀ U1, · · · , Uk ∈ T =⇒ ∩1≤i≤kUi ∈ T .

The sets in T are called the open sets of the topological space (X, T ).

(1) The usual topology TR on R:

U ∈ TR ⇐⇒ U = ∅ or union of open intervals (a, b).

(2) X = ∅ =⇒ trivial topology Ttrivial := ∅, X, and discrete topologyTdiscrete := 2X . Note that Ttrivial ⊂ Tdiscrete.

(3) Let T1, T2 be two topologies on X. We say T1 is coarser than T2 or T2 isfiner then T1, if T1 ⊂ T2. Then, for any topology T on a nonempty setX, we have Ttrivial ⊂ T ⊂ Tdiscrete.

Let (X, T ) be a topological space.

(1) A neighborhood of x (resp. of A) in X is a set N(x) (resp. N(A)) contain-ing an open set which contains x (resp. A).

Theorem 1.3.1. Let (X, T ) be a topological space. A subset A ⊂ X is open⇐⇒ it is aneighborhood of each of A.

♣ Exercise: Proof Theorem 1.3.1.


PROOF. If A is open, for any x ∈ A we can take N(x) = A. Conversely,suppose that A is a neighborhood of each of A. For each x ∈ A there is an openset Nx containing x and Nx ⊂ A. Then

A =∪

x∈ANx.

So A is open.

(2) x ∈ X is a limit point of A ⊂ X if ∀ neighborhood N(x) of x contains atleast one point a ∈ A different from x.

Theorem 1.3.2. A ⊂ X is closed⇐⇒ A contains all its limit points.


PROOF. Assume that A is closed and x is a limit point of A. If x /∈ A, thenx ∈ X \ A that is open. By Theorem 1.3.1 we conclude that X \ A is a neighborhoodof x. Because x is a limit point, the neighborhood X \ A must contain a point a ∈ A.It is impossible! So any limit points of A must be contained in A.

Now we assume that A contains all its limit points. To prove that A is closed,we suffice to show that X \ A is open. If X \ A is not open, then By Theorem 1.3.1X \ A is not a neighborhood of some x ∈ A. Hence any open set U containing x isnot contained in X \ A. Then U contains some a ∈ A so x is a limit point of A andhence x ∈ A. It is impossible!.

(3) The closure of A ⊂ X:

A := A ∪ limit points of A.(4) The support of f : X → K:

supp( f ) := x ∈ X : f (x) = 0.(5) The interior of A ⊂ X:

A := the largest open set contained in A.

(6) A is dense in X if A = X.(7) A is nowhere dense in X if A has an empty interior.(8) X is separable if it has a countable dense subsets.

1.3.2. Separation. A topological space (X, T ) is Hausdorff if any two distinctpoints possess disjoint neighborhoods.

(1) In a Hausdorff space the points are closed subsets.(2) The usual topology on R and the discrete topology are Hausdorff.(3) The trivial topology is not Hausdorff.(4) A topological space is normal if it is Hausdorff and if any disjoint closed

sets F1 and F2 have disjoint open neighborhoods U1 and U2.

1.3. TOPOLOGY 13

1.3.3. Base. A base B for a topology T is a subsystem of T which satisfieseither one the following equivalent conditions:

(i) ∀ element of T is the union of elements of B;(ii) ∀ x ∈ X and ∀ U ∈ T with x ∈ U ∃ B ∈ B such that x ∈ B ⊂ U.

Then T is said to be generated by B and is denoted TB .(1) A base for the usual topology of R:

B = (a, b) : a, b ∈ R.(2) A base for neighborhoods of x ∈ X is a family N of neighborhoods of x

such that any neighborhood of x contains a member of N .(3) The first countable space is a topological space in which each point has

a countable base of neighborhoods.(4) The second countable space is a topological space which topology is gen-

erated by a countable base B.

1.3.4. Convergence. A sequence (xn)n≥1 of points in a topological space (X, T )is a mapping N→ X given by n 7→ xn.

(1) A sequence (xn)n≥1 converges to x ∈ X if ∀ neighborhood N of x ∃ n0 ≥ 1such that N contains all xn (∀ n ≥ n0). Write limn→∞ xn = x or xn → x.

(2) If (X, T ) is a first countable space then the topology can be described interms of sequences.

Let (I,≤) be a directed set and (X, T ) a topological space.(1) A net (xα)α∈I in X is a mapping I → X given by α 7→ xα.(2) A net (xα)α∈I converges to x ∈ X if ∀ neighborhood N of x ∃ α0 ∈ I such

that xα ∈ N for all α ≥ α0.(3) Let (I′,≤′) be another directed set. A subnet (yα′)α′∈I′ of a net (xα)α∈I

is a mapping I′ → X given by α′ 7→ xϕ(α′) = yα′ , where ϕ : I′ → I isa mapping such that ∀ α ∈ I ∃ α′ ∈ I′ for which ϕ(β′) ≥ α wheneverα′ ≤′ β′.

1.3.5. Covering and compactness. A system (Ui)i∈I of (resp. open) subsets ofa topological space (X, T ) is a (resp. an open) covering if ∀ x ∈ X ∃ Ui such thatx ∈ Ui (i.e., ∪i∈IUi = X).

(1) U = (Ui)i∈I is finite if |I| < +∞.(2) A subcovering of the covering U is a subset of U which is itself a cover-

ing.(3) The covering V = (Vj)j∈J is a refinement of the covering U = (Ui)i∈I if∀ j ∈ J ∃ i ∈ I such that Vj ⊂ Ui.

(4) A covering U = (Ui)i∈I is locally finite if ∀ x ∈ X ∃ neighborhood Nsuch that N ∩Ui1 ∩ · · · ∩Uik = ∅ for some finite indices i1, · · · , ik ∈ I.

(5) A ⊂ X is compact if it is Hausdorff (relative to the subspace topology)and if any open covering of A has a finite open subcovering.

Theorem 1.3.3. (1) A compact subspace of a Hausdorff space is necessarily closed.(2) Any closed subspace of a compact space is compact.(3) (Bolzano-Weierstrass) A Hausdorff space is compact ⇐⇒ ∀ net has a conver-

gent subnet.


(4) In a metric space, a set A ⊂ X is compact ⇐⇒ ∀ sequence in A contains aconvergent subsequence with limit in A.

(5) (Heine-Borel theorem) The compact subsets of Rn are closed bounded subsetsof Rn.

The Heine-Borel theorem is in general not true.

A ⊂ X is relatively compact if A ⊂ X is compact. A space is locally compactif any point has a compact neighborhood. A Hausdorff space is paracompact ifany open covering has a locally finite refinement.

Theorem 1.3.4. All metric spaces are paracompact. In general, any locally compact,second countable, Hausdorff space is paracompact.

1.3.6. Connectedness. A topological space (X, T ) is disconnected if ∃ dis-joint nonempty open subsets A and B of X such that A ∪ B = X. Otherwise, X issaid to be connected.

Theorem 1.3.5. A topological space X is connected if and only if the only subsets whichare both open and closed are ∅ and X.

A topological space X is locally connected if any neighborhood of any x con-tains a connected neighborhood.

1.3.7. Continuous mappings. Let f : X → Y be a mapping between twotopological spaces (X, TX) and (Y, TY). We say that f is continuous at x ∈ X ifany neighborhoods B ⊂ Y of y = f (x) ∃ neighborhood A of x such that f (A) ⊂ B.

(1) f is continuous on X or is TX/TY-continuous if it is continuous at allx ∈ X.

Theorem 1.3.6. f : X → Y is continuous⇐⇒ ∀ open U in Y, V := f−1(U) is open inX.

Theorem 1.3.7. f : X → Y is continuous ⇐⇒ the net ( f (xα))α∈I converges to f (x)whenever the net (xα)α∈I converges to x.

♣ Exercise: Proof Theorem 1.3.6 and Theorem 1.3.7.

1.3. TOPOLOGY 15

PROOF. (1) Let U be open in Y and let V = f−1(U). Let x ∈ V. Then f (x) ∈U. Now U is a neighborhood of f (x). There exists N(x) such that f (N(x)) ⊂ U iff is continuous; so N(x) ⊂ V. Hence V is open.

Conversely, let W be a neighborhood of f (x). Then there is an open V in Wcontaining f (x) and f−1(V) is open in X. Since f−1(W) ⊃ f−1(V), it follows thatf−1(W) is a neighborhood of x such that f ( f−1(W)) ⊂W.

(2) Assume that f is continuous and the net (xα)α∈I converges to x. For anyneighborhood B ⊂ Y of f (x), by the continuity, there exists a neighborhood A ofx ∈ X such that f (A) ⊂ B. For this A, there is a α0 ∈ I such that xα ∈ A for allα ≥ α0. Because f (xα) ∈ B for all α ≥ α0, we see that the net ( f (xα))α∈I convergesto f (x).

Conversely, suppose that f is not continuous at x. There exists a neighborhoodB of f (x) such that for any neighborhood A of x we have f (A) * B. Hence we canfind a net (xα)α∈I that converges to x, but the corresponding net ( f (xα))α∈I doesnot converge to f (x), because f (xα) /∈ B.

(2) f is continuous =⇒ the sequence ( f (xn))n≥1 converges to f (x) wheneverthe sequence (xn)n≥1 converges to x.

(3) The converse of (2) is not true for non-countable spaces.

Theorem 1.3.8. (i) The image of a continuous mapping of a compact space is compact.(ii) Any continuous function on a compact space takes on its minimum and maximum.

(4) A continuous mapping f : X → Y is proper if f−1(A) is compact when-ever A is compact in Y.

(5) f : (X, TX)→ (Y, TY) is continuous =⇒ ∀ T ′X ⊃ TX and ∀ T ′Y ⊂ TY, wesee that f is also T ′X/T ′T-continuous.

(X, TX)f−−−−−−→

continuous(Y, TY)

finer

y ycoarser

(X, discrete topology)f−−−−−−→

continuous(Y, trivial topology)

(6) We can ask the following question:

(X, ??? the coarsest topology)f−−−−−−→

continuous(Y, ??? the finest topology)

Let X be a set and (Xα)α∈A a family of topological spaces.(6.1) The projective topology on X with respect to (Xα, fα)α∈A is the coars-

est topology on X for which each fα is continuous:

fα : X −→ Xα.

(6.2) The inductive topology on X with respect to (Xα, gα)α∈A is the finesttopology on X for which each gα is continuous:

gα : Xα −→ X.


(7) A homeomorphism is a bijection f which is bi-continuous (i.e., f and f−1

are continuous).

Theorem 1.3.9. A continuous bijection f : X → Y between two topological spaces is ahomeomorphism in the following cases:

(i) X and Y are compact, or(ii) (Banach) X and Y are Banach spaces and f is linear.

1.3.8. Simply-connectedness and path-connectedness. A covering space ofa topological space X is a pair (X, f ) where X is a connected and locally connectedspace, and f is a continuous mapping of X onto X such that ∀ x ∈ X ∃ neighbor-hood V of x satisfying that the restriction of f onto each connected component Cα

of f−1(V) is a homeomorphism from Cα onto V.

(1) Two covering spaces (X1, f1) and (X2, f2) are isomorphic if X1 is homeo-morphic to X2 by φ and f2 = f1 φ−1:

X1φ //

f1 AAA

AAAA

A X2

f2

X

(2) X is simply-connected if X is connected and locally connected, and anycovering space (X, f ) is isomorphic to the trivial covering space (X, IdX).

(3) R is simply-connected and (R, π) is a covering space of R := R/Z,where π(x) := x + Z.

(4) (X, f ) is an universal covering space for X if it is a covering space of Xand X is simply-connected.

(5) X is locally simply-connected if ∀ x ∈ X has a simply-connected neigh-borhood.

Theorem 1.3.10. (i) A connected and locally connected space has a universal coveringspace.

(ii) If a topological space X admits a universal covering space, it admits only one, upto isomorphisms.

(6) Let (X, f ) be a universal covering space of a topological space X. Thefundamental group of X is defined to be

π1(X) = φ : φ : X → X homeomorphic and f φ = f .

By Theorem 1.3.9 the fundamental group is independent of the choice ofX. Moreover, π1(X) is indeed a group.

(7) X is path-connected if ∀ a, b ∈ X ∃ continuous path between them, thatis, ∃ continuous mapping γ : [0, 1]→ X such that γ(0) = a and γ(1) = b.

1.3. TOPOLOGY 17

(8) X is locally path-connected if ∀ x ∈ X and ∀ neighborhood V of x ∃neighborhood U ⊂ X which is path-connected.

path-connected/locally path-connected =⇒ connected/locally connected.

However the converse is not true in general: the set

D :=(x, y) ∈ R2 : y = sin

1x

, x > 0

is locally path-connected, but it not path-connected.

♣ Exercise: Proof the above fact.

PROOF. Notice that

D =

(x, y) ∈ R2 : y = sin

1x

, x > 0∪ (0 × [−1, 1]) .

Then (0, 1) and (1, 0) can not be connected by path.

Let X be a topological space. Two paths γ0 and γ1 in X are homotopic if ∃continuous mapping

F : [0, 1]× [0, 1] −→ X, (t, s) 7−→ γs(t) := F(t, s)

such that F(t, 0) = γ0(t) and F(t, 1) = γ1(t).

Theorem 1.3.11. If X is a path-connected and locally path-connected space, and anyclosed path in X is homotopic to a constant, then X is simply-connected.

As a consequence, π1(Sn) is simply-connected for any n ≥ 2, but, π1(S

1) ∼= Z.The Poincare conjecture states that any closed simply-connected three-dimensionalmanifold is diffeomorphic to S3. This conjecture is completely solved now by us-ing the Ricci flow, where Hamilton, Perelman, and lots of people make great work.

1.3.9. Associated topologies. Let (X, T ), (X1, T1), (X2, T2) be topologicalspaces.

(1) The relative topology on A ⊂ X is

TA := A ∩U : U ∈ T .(2) (A, T ′A) is a topological space and A ⊂ X. We say that A ⊂ X is a

topological inclusion, written as A → X, if T ′A ⊃ TA.(3) A base for the product topology on X1 × X2 is U1 ×U2 : Ui ∈ Ti, 1 ≤

i ≤ 2. The α-th projection mapping is

πα : X1 × X2 −→ Xα, (x1, x2) 7−→ xα, 1 ≤ α ≤ 2.

The product topology is the coarsest topology in which each πα is contin-uous.

(4) All the open balls on Rn form a base for the usual topology on Rn that isequivalent to the product topology on R× · · · ×R (n times).

Let (Xα)α∈A be a family of topological spaces.


(1) Cartesian product:

∏α∈A

Xα :=

x : A −→

∪α∈A

Xα : xα = x(α) ∈ Xα

.

(2) Letπα : ∏

α∈AXα −→ Xα, x 7−→ xα.

(3) The product topology on ∏α∈A Xα is the coarsest topology in which πα

is continuous. A base for the product topology consists of

∏1≤i≤n

Uαi × ∏α =i1,··· ,in

Xα, Uαi ∈ Tαi .

(4) Tychonoff’ theorem: The arbitrary product of compact spaces is compactwith respect to the product topology.

1.3.10. Topology related on other structures. A set X together with a groupoperation an a topology is said to be a topological group if

X× X −→ X, (x, y) 7−→ xy and X −→ X, x 7−→ x−1

are both continuous.

A topological space X which is also a vector space on K is said to be a topo-logical vector space if

X× X −→ X, (x, y) 7−→ x + y and K× X −→ X, (λ, x) 7−→ λx

are both continuous.(1) Rn together with its usual topology is a topological vector space.(2) A topological vector space is said to be locally convex if it admits a base

of neighborhood of 0 (zero element) made of convex sets.

Theorem 1.3.12. Let T : X → Y be a linear mapping between two topological spaces.Then T is continuous if and only if T is continuous at 0 ∈ X.

(3) If X and Y are topological vector spaces, then we denote by L(X, Y) theset of all linear continuous mappings from X to Y, and

(1.3.1) X′ ≡ topological dual of X := L(X, K).

For any x′ ∈ X′ write x′(x) = ⟨x′, x⟩ ∈ K for any x ∈ X.(4) A bounded set in a topological vector space is a set which can be mapped

inside any neighborhood of the origin by a homothetic transformation ofsufficiently small ratio centered on the origin (x 7→ ϵx).

1.3.11. Metric spaces. A metric space is a pair (X, d), where X is a set andd : X× X → R is a mapping, satisfying

(i) d(x, y) ≥ 0,(ii) d(x, y) = 0⇐⇒ x = y,

(iii) d(x, y) = d(y, x),(iv) d(x, z) ≤ d(x, y) + d(y, z).

We call d(x, y) the distance between x and y.

1.3. TOPOLOGY 19

(1) Open balls:

B(x, r) := y ∈ X : d(x, y) < r.

Theorem 1.3.13. (X, d) is a metric space =⇒ TB is a topology on X, where B =B(x, r) : x ∈ X, r > 0.

(2) TB is called the topology induced by the metric d.(3) (X, d) is a metric space =⇒ X is first countable, Hausdorff and normal.(4) Examples.

(4.1) (X, d0) is a metric space =⇒ discrete topology, where d0(x, y) = 0 ifx = y, and 1 if x = y.

(4.2) X = R2, define

d1(x, y) := [(x1 − y1)2 + (x2 − y2)2]1/2,

d2(x, y) := max|x1 − y1|, |x2 − y2|,d3(x, y) := |x1 − y1|+ |x2 − y2|.

(4.3) (Rn, d) is a metric space, with

d(x, y) :=

(∑

1≤i≤n(xi − yi)2

)1/2

.

Euclidean topology on Rn induced by d is equivalent to the producttopology on R× · · · ×R (n times).

(4.4) Let (X, d) be a metric space. For x, y ∈ X define

Path(x, y) := γ : [0, 1]→ X|γ continuous and γ(0) = x, γ(1) = y.

Thendx,y(γ1, γ2) := sup

0≤t≤1d(γ1(t), γ2(t))

is a metric on Path(x, y).(4.5) Let (X, d) be a metric space and γ : [0, 1]→ X a path. Define

[γ] :=

γ′ : [0, 1]→ X continuous∣∣∣ γ′ = γ φ, φ : [0, 1]→ [0, 1]

monotonic and continuous

.

Then (X, d) is a metric space, where

X := [γ]|γ : [0, 1]→ X continuous

and

d([γ1], [γ2]) := infγ′i∈[γi ], 1≤i≤2

(sup

0≤t≤1d(γ′1(t), γ′2(t))

).

♣ Exercise: Verify (4.1) – (4.5).


PROOF. (4.1): By definition

B(x, r) =x, r ∈ (0, 1],X, r ∈ (1,+∞).

Hence d0 induces the discrete topology on X.(4.2): trivial.(4.3): trivial.(4.4): Obviously dx,y(γ1, γ2) ≥ 0 and dx,ty(γ1, γ2) = dx,y(γ2, γ1). The triangle

inequality for dx,y follows from that inequality for d.(4.5): the same as (4.4).

Theorem 1.3.14. Let f : X → Y be a mapping from a metric space (X, d) to a topologicalspace (Y, T ). Then f is continuous at x if and only if ( f (xn))n≥1 converges to f (x)whenever (xn)n≥1 converges to x.

(5) A Cauchy sequence in a metric space (X, d) is a sequence (xn)n≥1 suchthat limn→∞ d(xn, xm) = 0.

(6) A metric space is complete if any Cauchy sequence is convergent.(7) Any metric space is dense in a complete metric space.

A topological invariant is a property of topological spaces which is preservedunder a homeomorphism. For example, separation properties, compactness, con-nectedness, etc.

Exercise 1.3.15. Show that the completeness is not a topologically invariant prop-erty.

PROOF. Consider the homeomorphic mapping

f : [0,+∞) −→ [0, 1), x 7−→ x1 + x

.

However, [0,+∞) is complete while [0, 1) is not.

A topological vector space X is said to be metrizable if its topology can beinduced by some metric d invariant by translation, i.e.,

d(x, y) = d(x + z, y + z), ∀ x, y, z ∈ X.

A Frechet space is a complete, metrizable, topological vector space.

1.3.12. Banach spaces. Let X be a vector space over K. A norm on X is amapping || · || : X → R such that

(i) ||x + y|| ≤ ||x||+ ||y||,(ii) ||λx|| = |λ|||x||,

(iii) ||x|| = 0⇐⇒ x = 0.

1.3. TOPOLOGY 21

Then ||0|| = 0, ||x|| ≥ 0, and

||x− y|| ≥∣∣∣∣||x|| − ||y||∣∣∣∣.

♣ Exercise: Please check it!

PROOF. Compute

||0|| = ||0x|| = |0| · ||x|| = 0,

0 = ||x− x|| ≤ ||x||+ || − x|| = 2||x|| =⇒ ||x|| ≥ 0.Finally,

||x|| = ||(x− y) + y|| ≤ ||x− y||+ ||y|| =⇒ ||x− y|| ≥ ||x|| − ||y||and ||x− y|| = | − 1| · ||y− x|| ≥ ||y|| − ||x||.

(1) || · || is a seminorm if it satisfies (i) and (2).(2) A norm || · || induces a metric on X which is invariant by translation:

||x− y|| := d(x, y).

(3) A normed vector space is a pair (X, || · ||) where X is a vector space and|| · || is a norm on X. Therefore

(X, || · ||) −−−−→ (X, d) −−−−→ (X, T )∥∥∥ ∥∥∥ ∥∥∥normed vector space metric space topological space

(4) || · || and || · ||′ are norms on X such that ||x| < ||x||′ for all x ∈ X =⇒B′(x, ϵ) ⊂ B(x, ϵ) and T ⊂ T ′.

(5) || · || and || · ||′ are norms on X and T ⊂ T ′ =⇒ ∃ λ > 0 such that||x|| < λ||x||′ for all x ∈ X.

Indeed, the open ball Bϵ := B(0, ϵ) contains B′η := B′(0, η) =⇒ ηx/2||x||′ ∈B′η ⊂ Bϵ =⇒ ||ηx|| < 2ϵ||x||′ =⇒ λ = 2ϵ/η > 0.

Proposition 1.3.16. Any normed vector space must be a metrizable topological vectorspace.

♣ Exercise: Proof Proposition 1.3.16.

PROOF. Suppose that (X, || · ||) is a normed vector space. We shall prove that

(x, y) 7→ x + y and (λ, x) 7→ λx are both continuous.

Indeed,||(x + y)− (x0 + y0)|| ≤ ||x− x0||+ ||y− y0||

and||αx− α0X0|| = ||α(x− x0) + (α− α0)x0||


≤ |α| · ||x− x0||+ |α− α0| · ||x0||≤ |α− α0| (||x− x0||+ ||x0||) + |α0| · ||x− x0||.

Proposition 1.3.17. Any normed vector space must be a locally convex topological vectorspace.

PROOF. A base B = B(x, r) : x ∈ X, r > 0 for (X, || · ||). We shall provethat B(x, r) is convex. ∀ x1, x2 ∈ B(x, r) and λ ∈ [0, 1], ||λx1 + (1− λ)x2 − x|| ≤λr + (1− λ)r = r.

(6) A seminormed vector space is a pair (X, || · ||) where X is a vector spaceand || · || is a seminorm. It is clear that Proposition 1.3.17 is also true forany seminormed vector spaces.

(7) A Banach space is a complete normed vector space. According to Propo-sition 1.3.16 any Banach space is Frechet.

Example 1.3.18. (1) (Rn, || · ||) is complete, where

||x|| := d(0, x) =

(∑

1≤i≤n(xi)2

)1/2

.

(2) (CB(X), || · ||CB(X)) is complete, where(1.3.2)

CB(X) := f : X → R continuous and bounded, || f ||CB(X) := supx∈X| f (x)|.

(3) If U ⊂ Rn is open, then (CkB(U), || · ||Ck

B(U)) is complete, where

CkB(U) :=

f : U → R

∣∣∣∣ continuous and uniformly boundedderivatives of order ≤ k

,

and

(1.3.3) || f ||CkB(U) := sup

x∈U∑|α|≤k|Dα f (x)|

with

α := (α1, · · · , αn) ∈Nn, |α| = ∑1≤i≤n

αi, Dα f (x) := ∏1≤i≤n

(∂

∂xi

)αi

f (x).

(4) If U ⊂ Rn is open, then (L2(U), || · ||L2(U)) is not complete, where

(1.3.4) L2(U) := f : U → R continuous and square integrableand

|| f ||L2(U) :=[∫

U| f (x)|2dx

]1/2.

1.3. TOPOLOGY 23

(8) Compact subsets of a metric space are also closed and bounded.

Let (X, || · ||) be a normed space. A sequence ( fn)n≥1 converges strongly tof ∈ X if

limn→∞

|| f − fn|| = 0.

Consequently, limn→∞ || fn|| = || f ||.

Let (X, T ) be a topological vector space, and X′ := L(X, K) be its topologicaldual.

(1) ( fn)n≥1 converges weakly to f ∈ X, written as fn f , if

limn→∞⟨g, fn⟩ = ⟨g, f ⟩, ∀ g ∈ X′.

(2) If T is induced from a norm || · || on X, then (over R)(2.1) fn f =⇒ || f || ≤ lim infn→∞ || fn||,(2.2) fn f and limn→∞ || fn|| = || f || =⇒ limn→∞ || fn − f || = 0.

(3) ( fn)n≥1 ⊂ X′ converges ∗-weakly to f ∈ X′, written as fn ∗ f , if

limn→∞⟨ fn, g⟩ = ⟨ f , g⟩, ∀ g ∈ X.

(4) If X is reflextive (i.e., X′ = X), then weak topology is equivalent to ∗-weak topology.

Theorem 1.3.19. All bounded subsets are relatively compact in the weak topology of areflextive Banach space.

1.3.13. Hilbert spaces. A Hilbert space is a complete pre-Hilbert space (H , || ·||), where ||x|| := (x|x)1/2.

(1) A Hilbert space is real if the underlying vector space is defined on R.

Theorem 1.3.20. (Riesz’s representative theorem) There is an isomorphism betweena Hilbert space H and its topological dual H ′ defined by

H −→H ′, x 7−→ (x|·).

(2) A subset (xα)α∈A of a Hilbert space H is called an orthonormal base ofH if it is a base such that

(xα|xβ) = δαβ, ∀ α, β ∈ A.

(3) Any Hilbert space H admits an orthonormal base.(3.1) Two orthonormal bases have the same cardinality.(3.2) Two Hilbert spaces with orthonormal bases of the same cardinality

are isomorphic.(4) Any separable Hilbert space is isomorphic to L2((0, 1)).


(5) For any element x of a separable Hilbert space we have

x = ∑n≥1

(x|xn)xn,

where (xn)n≥1 is a countable base of H .

1.4. Measures and integrations

1.4.1. Measures. Let X be a nonempty set.

(1) A ring of subsets of X is a nonempty class R of subsets of X such that

A, B ∈ R =⇒ A \ B, A ∪ B ∈ R.

Hence ϕ ∈ R and R is closed under finite unions and finite intersections.(2) A ring of subsets R is a ring under

A + B := AB = (A \ B) ∪ (B \ A) ∈ R,AB := A ∩ B = (A ∪ B) \ (AB) ∈ R.

Moreover the zero element and the unit element are respectively ∅ andX.

(3) The ring R is called a field if X ∈ R. It is then denoted A .(4) The ring R (resp. the field A ) is called a σ-ring (resp. σ-field) if R (resp.

A ) is closed under countable unions:

Ai ∈ Ri (i ≥ 1) =⇒∪i≥1

Ai ∈ R.

(5) A σ-field A can equally defined by the following axioms:

A ∈ A =⇒ X \ A ∈ A , Ai ∈ A =⇒∪i≥1

Ai ∈ A .

(6) Let E be a class of subsets of X =⇒ ∃! σ-field A (E ), called the σ-fieldgenerated by E , containing E :

(1.4.1) A (E ) := the smallest class of subsets of Xcontaining E such that A (E ) is a σ-field.

The existence is guaranteed by Zorn’s lemma.(7) A measurable space is a pair (X, A ), where X is a set and A is a σ-field.

Let (X, T ) be a topological space.

(1) The Borel σ-field of X is the σ-field generated by the open sets of X (orequivalently the closed sets of X). An element of the Borel σ-field is calleda Borel set.

(2) ∪i≥1Ui, ∪i≥1Fi, ∩i≥1Ui, ∩i≥1Fi are Borel sets, where Ui and Fi are respec-tively open and closed sets.

(3) Gδ-set: ∩i≥1Ui (Ui open), Fσ-set: ∪i≥1Fi (Fi closed), Gδσ-set: ∪i≥1Gi (Giare Gδ-sets), Fσδ-set: ∩i≥1Fi (Fi are Fσ-sets).

1.4. MEASURES AND INTEGRATIONS 25

(4) The Borel σ-field of R is generated by the open intervals (a, b) or equiv-alently by the closed intervals [a, b], or the semi-closed [a, b) or (a, b] or(−∞, a] or (−∞, a) or [a,+∞) or (a,+∞).

Indeed, an open set is a countable union of open intervals. For exam-ple

(a, b) =∪

n≥1

[a +

1n

, b)

, [a, b) =∪

n≥1

[a, b− 1

n

].

1.4.2. Measure spaces. Let (X, T ) be s topological space.(1) A positive set function on X is a mapping m : A → [0,+∞], where

ϕ ∈ A ⊂ 2X .(1.1) m is finitely additive if ∀ disjoint finite family (Ai)1≤i≤n of A with

∪1≤i≤n Ai ∈ A ,

m

( ∪1≤i≤n

Ai

)= ∑

1≤i≤nm(Ai), m(∅) = 0.

(1.2) m is countably additive if ∀ disjoint countable family (Ai)i≥1 of Awith ∪i≥1 Ai ∈ A ,

m

(∪i≥1

Ai

)= ∑

i≥1m(Ai), m(∅) = 0.

(1.3) m is σ-finite if X = ∪i≥1 Ai with m(Ai) < +∞.(1.4) m is finite if X ∈ A and m(X) < +∞.

(2) Let (X, A ) be a measurable space.(2.1) A positive measure m on (X, A ) is a countably additive, positive set

function m : A → [0,+∞].(2.2) A measure space is a triple (X, A ,m), where (X, A ) is a measurable

space and m is a positive measure on (X, A ).The elements of A are the measurable subsets of X, and m(A) is themeasure of A ∈ A .

Theorem 1.4.1. (Hahn extension theorem) Let X be a set, Σ a field of subsets of X,and µ : Σ → [0,+∞] a countably additive positive set function. Then there is a positivemeasure µ on the σ-field generated by Σ such that µ Σ = µ and µ is unique when µ isσ-finite.

Σµ //

inclusion

[0,+∞]

A (Σ)µ

::uuuuuuuuu

(3) A probability space is a measure space (X, A ,m) with m(X) = 1. Theelements A of A are called events and m(A) is the probability of A.

(4) A property is said to hold m-almost everywhere (m-a.e.) if it holds for allpoints of X except possibly for points of a set A of measure m(A) = 0.


(5) A positive measure m in the measure space (X, A ,m) is said to be com-plete if ∀ A ∈ A with m(A) = 0 and ∀ B ⊂ A, we have B ∈ A andm(B) = 0.

Any measure space (X, A ,m) can be included into a complete mea-sure space (X, A ,m).

(6) A positive measure m on (X, B), where X is a locally compact Hausdorffspace and B is the Borel σ-field, is called a Borel measure if the measureof every compact set (so is closed and hence measurable) is bounded.

(7) Let Σ be a field of subsets of X and µ : Σ → [0,+∞] be finitely additive,positive set function. We say that µ is regular if ∀ A ∈ Σ and ∀ ϵ > 0 ∃F, G ∈ Σ such that

F ⊂ A ⊂ G and µ(G \ F) < ϵ.

Theorem 1.4.2. (Alexandroff theorem) Let X be compact topological space, Σ a fieldof subsets of X. If µ is a finite, regular, finitely additive, positive set function on Σ, then µis countably finite.

(9) A Borel measure µ is regular⇐⇒ ∀ Borel set A and ∀ ϵ > 0 ∃ closed setF and open set U such that F ⊂ A ⊂ U and µ(U \ F) < ϵ.

Let f : X → Y be a set mapping.(1) Given a metric space (X, A ,m), define the image by f of (X, A ,m) to be

the metric space (Y, B, n), where

B := B ∈ 2Y : f−1(B) ∈ A , n(B) := m( f−1(B)) (∀ B ∈ B).

(2) A positive measure (A ,m) on X is invariant by f : X → X if

f−1(A) ∈ A (∀ A ∈ A ) and m(A) = m( f−1(A)).

Let M = (X, A ,m) and M′ = (X′, A ′,m′) be two σ-finite measure spaces.(1) Define the σ-field A ⊗A ′ of subsets of X× X′ by

A ⊗A ′ := the smallest σ-field containing all the sets ofthe form A× A′, where A ∈ A and A′ ∈ A ′

(2) By Theorem 1.4.1, ∃! positive measure m⊗ m′ defined on A ⊗A ′ suchthat

(m⊗m′)(A× A′) = m(A)m′(A′), ∀ (A, A′) ∈ A×A ′.

(3) The product measure space:

M×M′ = (X× X′, A ⊗A ′,m⊗m′).

The Lebesgue measure on Rn.


(1) The Lebesgue measure on R:

(R, B, l),

where R is the real line with its usual topology, B is the Borel σ-field gen-erated by open intervals, and l, called the Lebesgue measure on R, is theonly regular Borel measure, up to multiplication by constant, invariantby translation.

(2) Define l as follows:(2.1) l((a, b)) := b − a, ∀ open interval (a, b) ∈ R (=⇒ l is an additive

positive set function).(2.2) Extend l, defined in (2.1), to the field generated by open intervals

=⇒l([a, b]) = l([a, b)) = l((a, b]) = b− a.

(2.3) If [a, b) = ∪i≥1[ai, bi), where [ai, bi) are pairwise disjoint, then

l([a, b)) = ∑i≥1

l([ai, bi)).

(3) Cantor set:C := [0, 1] \ S,

where

S :=∪i≥1

Si, S1 =

(13

,23

), S2 =

(19

,29

)∪(

79

,89

), · · · .

Then C is uncountable, but is measure zero with respect to l.(4) The Lebesgue measure on Rn:

Rn :=

R× · · · ×R︸︷︷︸n

, B ⊗ · · · ⊗B︸︷︷︸n

, l⊗ · · · ⊗ l︸︷︷︸n

.

Let G be a locally compact topological group and B its Borel σ-field.(1) A (left) Haar measure µ is a left-invariant (i.e., invariant under the map-

pings G → G, x 7→ gx, that is, µ(gA) = µ(A) for all Borel sets A and allg ∈ G), regular, Borel measure on G. Similarly, we can define right Haarmeasure µ (that is µ(Ag = µ(A)).

Theorem 1.4.3. There is a unique Haar measure, modulo a constant factor, on a givenlocally compact topological group.

(2) Let G = R∗ = R \ 0 with group law the usual multiplication. Thendx/|x| is a Haar measure:∫

Rf (tx)

dx|x| =

∫R

f (x)dx|x|

for all f ∈ L1(G, l).



PROOF. If t > 0, then considering y := tx yields∫R

f (tx)dx|x| =

∫ +∞

−∞f (y)

dy|y| .

If t < 0, then considering z := tx yields∫R

f (tx)dx|x| =

∫ −∞

+∞f (y)

dy−|y| =

∫ +∞

−∞f (x)

dx|x| .

1.4.3. Measurable functions. Let f : X → Y be a mapping between two mea-surable spaces (X, A ) and (Y, B).

(1) f is measurable or A /B-measurable if f−1(B) ∈ A for all B ∈ B.(2) Examples:

(2.1) If (Y, B, n) is the image of (X, A ,m) by f : X → Y, then f is measur-able.

(2.2) If (X, A ) and (Y , B) are topological spaces with Borel σ-fields, thenany continuous mapping from X to Y is measurable.

(3) The composition of two measurable mappings is also measurable:

(X, A )f−−−−−−→

measurable(Y, B)

g−−−−−−→measurable

(Z, C ) =⇒ g f measurable.

Then we obtain a category MS:

Ob(MS) : measurable spaces,

and

HomMS((X, A ), (Y, B)) = measurable mappings.

(4) Let (X, A ,m) be a measure space and f : X → R a real function. Wesay f is measurable if it is a measurable mapping from (X, A ) to R, orequivalently, if x ∈ X : a < f (x) < b ∈ A , ∀ a < b.

(4.1) A complex-valued function f +√−1g is measurable if both f and g

are measurable.(4.2) f and g are measurable functions on X and λ ∈ K =⇒ λ f , f + g,

f − g, | f | are measurable.(5) (X, A ,m) is a complete measurable space, f = g is m-a.e. =⇒ f is mea-

surable if and only if g is measurable.

Theorem 1.4.4. (Lusin) Let X be a locally compact topological space with X = ∪i≥1Xi(Xi being compact), and m a Borel measure on X. TFAE:

(i) a function f on X is measurable,


(ii) ∀ compact set K ⊂ X and ∀ ϵ > 0 ∃ compact subset Kϵ ⊂ K and gϵ ∈ C(Kϵ, R)such that

gϵ = f |Kϵ and m(K \ Kϵ) < ϵ.This function gϵ may be extended to a continuous function g with compact sup-port in X, and supX | f | = supX |g|.

Let (X, A ,m) be a measure space.

(1) If ( fn)n≥1 is a sequence of measurable functions on (X, A ,m) that arefinite m-a.e., then we say that ( fn)n≥1 converge in measure to the mea-surable function f , if ∀ ϵ > 0 one has

limn→+∞

m (x ∈ X : | fn(x)− f (x)| > ϵ) = 0.

Notion: fn →m f .(2) We have the following relations:

convergence in measure a.e.

×y×

pointwise convergence a.e.

convergence in measure a.e.xµ(X)<+∞

pointwise convergence a.e.

Theorem 1.4.5. (Egorov) Let (X, A ,m) be a measure space with m(X) < +∞. If( fn)n≥1 converges pointwise a.e. to a finite measurable function f , then ∀ ϵ > 0 ∃Aϵ ⊂ X such that m(X \ Aϵ) < ϵ and ( fn)n≥1 uniformly converges to f on Aϵ.

1.4.4. Integrable functions. Let (X, A ,m) be a measure space.

(1) A function on (X, A ,m) is said to be simple if it is zero except on a finitenumber n of disjoint sets Ai ∈ A of finite measure m(Ai), where thefunctions is equal to a finite constant ki (1 ≤ i ≤ n).

(2) If f is a simple function, we define

(1.4.2)∫

Xf dm := ∑

1≤i≤nkim(Ai).

(3) Any measurable function f : X → R := R ∪ ±∞ ∃ ( fn)n≥1 such thatfn are simple functions and fn → f pointwise. If moreover f ≥ 0, thenthe fn may be chosen positive and ( fn)n≥1 increasing

0 ≤ f1 ≤ · · · ≤ fn ≤ fn+1 −→ f .

(4) For any measurable function f : X → [0,+∞], define

(1.4.3)∫

Xf dm := sup

∫X

ρdm : 0 ≤ ρ ≤ f , ρ simple

.

We then say that f is integrable if∫

X f dm is finite.


Theorem 1.4.6. (Monotone convergence theorem) Let (X, A ,m) be a measure spaceand ( fn : X → [0,+∞])n≥1 an increasing integrable sequence which is convergent m-a.e.to f . Then

limn→∞

∫X

fn dm exists =⇒ f is integrable and∫

Xf dm = lim

n→∞

∫X

fn dm.

Theorem 1.4.7. (Fatou) Let (X, A ,m) be a measure space and ( fn : X → [0,+∞])n≥1an measurable sequence which is convergent m-a.e. to f . Then

(1.4.4)∫

Xf dm ≤ lim inf

n→∞

∫X

fn dm.

(5) For ∀ f : X → R we have

f = f+ − f−

with f± : X → [0,+∞], and say f+ (resp. f−) the positive part (resp.negative part) of f . We say f is integrable if f± are both integrable. Itsintegral is defined to be∫

Xf dm :=

∫X

f+ dm−∫

Xf− dm.

Theorem 1.4.8. Let (X, A ,m) be a measure space. A measurable function f : X → R isintegrable if and only if | f | : X → [0,+∞] is integrable.

(6) Integration on A ∈ A . If A = ∪i≥1 Ai ∈ A with (Ai)i≥1 disjoint, then

(1.4.5)∫

Af dm =

∫X

f χA dm = ∑i≥1

∫Ai

f dm

for any measurable function f : A→ R.(7) Basic properties:

(7.1) For any λ, µ ∈ R,∫A(λ f + µg)dm = λ

∫A

f dm+ µ∫

Agdm.

(7.2) | f | ≤ |g|, g integrable, f measurable =⇒ f integrable.(7.3) f ≤ g m-a.e., f and g integrable =⇒∫

Af dm ≤

∫A

gdm.

(7.4) f measurable, | f | bounded on A ∈ A with m(A) < +∞ =⇒ f isintegrable and ∣∣∣∣∫A

f dm∣∣∣∣ ≤ Mm(A)

where | f | ≤ M on A.


(7.5) f ≥ 0, A ⊂ B and A, B ∈ A =⇒∫A

f dm ≤∫

Bf dm.

Theorem 1.4.9. (Lebesgue dominated convergence theorem) Let (X, A ,m) be ameasure space and ( fn)n≥1 an integrable sequence, fn → f a.e., | fn| ≤ g with g beingintegrable. Then f is integrable and

(1.4.6) limn→∞

∫X

fn dm =∫

Xlim

n→∞fn dm =

∫X

f dm.

(9) For (X, A ,m) = (X1 × X2, A1 ⊗A2,m1 ⊗m2), define∫X1×X2

f dm1dm2 :=∫

Xf dm.

Theorem 1.4.10. (Fubini) A measurable function f : X1 × X2 → R is integrable⇐⇒one of the following integrals∫

X1

[∫X2

| f |dm2

]dm1 and

∫X2

[∫X1

| f |dm1

]dm2

exists and is finite.If f is integrable, then

(1.4.7)∫

X1×X2

f dm1dm2 =∫

X1

[∫X2

f dm2

]dm1 =

∫X2

[∫X1

f dm1

]dm2.

(10) Let (Y, B, n) be an image under u of a measure space (X, A ,m).(10.1) A measurable function f on Y is integrable on Y ⇐⇒ f u is inte-

grable on X. Then

(1.4.8)∫

Yf dn =

∫X

f udm =∫

X(u∗ f )dm.

(10.2) f is measurable on Y =⇒ f u is measurable on X.(11) Let (X, A ,m) be a measure space and Y ∈ A . Then (Y, A ,m) is a mea-

sure space, where

A :=

A = Y ∩ A : A ∈ A

, m(A) := m(A).

We say (A ,m) the measure induced by m on Y.If ι : Y → X denotes the inclusion, then for any measurable function

f on X such that– χY f is measurable on X,– χY f is m-integrable,– f ι is m-integrable,

we obtain ∫Y

f ιdm =∫

XχY f dm =

∫Y

f dm.


1.4.5. Integration on locally compact spaces. Let (X, B,m) be a measure space,where X is a locally compact topological space, B is the Borel σ-field, and m is aBorel measure.

(1) m(compact set) is finite.(2) ∀ continuous function f with compact support =⇒∫

Xf dm is finite.

(3) The space of continuous functions with compact support on X is densein the space of integrable functions on X.

Lebesgue integral.(1) (Rn, BRn , ln) =⇒ dx := dln and∫

Rnf dln =

∫Rn

f (x)dx.

Theorem 1.4.11. (Change of variable) Suppose that f is Lebesgue integrable on anopen subset V ⊂ Rn and φ : U → V, x 7→ y = φ(x), is a diffeomorphism. Then

(1.4.9)∫

Vf (y)dy =

∫U( f φ)(x)

∣∣det[φ′(u)]∣∣ dx

where φ′(x) = (∂yi/∂xj) = D(y)/D(x).

(2) Examples:(2.1) ∃ Lebesgue integrable but NO Riemann integrable:

f : [0, 1] −→ R, x 7−→ f (x) =

1, x ∈ Q,0, x /∈ Q.

(2.2) ∃ functions on R which are not Lebesgue integrable but have finiteimproper Riemann integrals (a measurable function f is Lebesgueintegrable if and only if | f | is Lebesgue integrable):

sin xx

, cos(x2), sin(x2), · · · .

(2.3) @ reasonable definition of improper Riemann integrals on Rn:

I :=∫∫

x,y>0sin(x2 + y2)dxdy.

The domain D = (x, y) ∈ R2 : x, y > 0 can be approximated byD′n := [0, n]× [0, n] or D′′n := x2 + y2 ≤ n2 : x, y > 0. However

I′n :=∫∫

D′nsin(x2 + y2)dxdy→ π

4, I′′n =

∫∫D′′n

sin(x2 + y2)dxdy =1− cos(n2)

2.

Radom measure. Let X be a locally compact space.


(1) A Radom measure µ on X is a continuous linear form

µ : D0(X) −→ R

where D0(X) is the space of continuous functions with compact supporton X endowed with the inductive limit topology of the topologies of uni-form convergence on compact sets.

(2) µ is continuous⇐⇒ ∀ compact set K ⊂ X ∃ C(K) > 0 such that ∀ contin-uous function f with support in K,

|µ( f )| ≤ C(K) supK| f |.

(3) A Radom measure µ is positive if µ( f ) ≥ 0 for all 0 ≤ f ∈ D0(X).(4) To any Borel measure m on X, we can associate a positive Radom measure

µ defined by

(1.4.10) µ( f ) :=∫

Xf dm, f ∈ D0(X).

Theorem 1.4.12. (Riesz-Markov) Suppose that X is a locally compact space and X =∪i≥1Xi with Xi being compact. Then ∀ positive Radom measure µ on X ∃! regular Borealmeasure m such that (1.4.10) holds for all f ∈ D0(X).

1.4.6. Signed and complex measures. A signed measure space is a triple(X, A ,m), where (X, A ) is a measurable space and m : A → (−∞,+∞] is acountably additive set function.

Theorem 1.4.13. (Jordan decomposition theorem) Given a signed measure space(X, A ,m), there is a unique decomposition

m = m+ −m−

where• m± are positive measures.• ∃ A ∈ A such that m−(A) = 0 and m+(X \ A) = 0.

The positive measure (A , |m|) with |m| := m+ + m− is called the total variation of(A ,m).

(1) f is integrable with respect to m⇐⇒ f is integrable with respect to |m|.(2) f is integral with respect to m =⇒∫

Xf dm :=

∫X

f dm+ −∫

Xf dm−,

∣∣∣∣∫Xf dm

∣∣∣∣ ≤ ∫X| f |d|m|.

A complex measure space is a triple (X, A ,m), where (X, A ) is a measurablespace and m : A → (−∞,+∞] +

√−1(−∞,+∞] is a set function with m = m1 +√

−1m2 and mi being signed measures.


(1) The total variation (A , |m|) of (A ,m) is defined to be(1.4.11)

|m|(A) := sup

∑i≥1|m(Ai)| : (Ai)i≥1 pairwise disjoint in A and ∪i≥1 Ai ⊆ A

.

Then

(1.4.12) |m(A)| ≤ |m|(A).

(2) If m is a signed measure, then |m|(A) defined in (1) agrees with its defi-nition in terms of the Jordan decomposition.

(3) Let f : X → C, f = f1 +√−1 f2, and f1, f2 real. We say that f is inte-

grable with respect to the complex measure m = m1 +√−1m2, if fi is

integrable with respect to mi (1 ≤ i ≤ 2). Its integral is∫X

f dm :=∫

Xf1 dm1 −

∫X

f2 dm2 +√−1(∫

Xf1 dm2 +

∫X

f2 dm1

).

(4) f is m-integrable⇐⇒ | f | is |m|-integrable and f is measurable. Moreover∣∣∣∣∫Xf dm

∣∣∣∣ ≤ ∫X| f |d|m|.

1.4.7. Integration of vector-valued functions. Let f be a mapping from ameasure space (X, A ,m) to a Banach space (E, B, || · ||).

(1) The first difficulty: f and g measurable ; f ± g is measurable.(2) The second difficulty: f measurable ; ∃ ( fn)n≥1 simply functions con-

verging pointwise to f .(3) When (X, A ,m) is a σ-finite measure space and (E, B, || · ||) is a separable

Banach space, the above two difficulties do not arise.(4) We say f is a simple mapping if it has value zero except on a finite

number Ai (1 ≤ i ≤ n) of subsets Ai ∈ A of X, with finite measurem(Ai) < +∞, where it has a constant value in E:

f = ∑1≤i≤n

aiχAi , ai ∈ E.

(5) For any simple mapping f , define∫X

f dm := ∑1≤i≤n

aim(Ai) ∈ E.

(6) A Cauchy sequence of simple mappings is a sequence ( fn)n≥1 of simplemappings such that ∀ ϵ > 0 ∃ N ∈N such that∫

X|| fn − fk||dm < ϵ

for all n, k > N.(6.1) ∀ n ≥ 1, one has∣∣∣∣∣∣∣∣∫X

fn dm∣∣∣∣∣∣∣∣ ≤ ∑

1≤i≤Mn

||ai,n||m(Ai,n) ≤∫

X|| fn||dm.

(6.2) ( fn)n≥1 Cauchy sequence of simple mappings =⇒ (∫

X fndm)n≥1 is aCauchy sequence in E =⇒ limn→∞

∫X fndm exists in E.


(7) f : X → E is m-integrable if ∃ Cauchy sequence ( fn)n≥1 of simple map-pings, converging a.e. to f . The integral of f is∫

Xf dm := lim

n→∞

∫X

fn dm

which is independent of the choice of ( fn)n≥1.

Theorem 1.4.14. Let f : (X, A ,m) → (B, B, || · ||) be a measurable mapping from ameasure space into a Banach space. Then

f is m-integrable ⇐⇒ || f || : X → [0,+∞) is integrable.

Then ∣∣∣∣∣∣∣∣∫Xf dm

∣∣∣∣∣∣∣∣ ≤ ∫X|| f ||dm.

1.4.8. L1-spaces. Let (X, A ,m) be a measure space.(1) Define

L 1(X) := integrable functions over X

=

f : X → R∪ +∞

∣∣∣∣ f measurable and∫X | f |dm < +∞

.(1.4.13)

(2) Define

L 1(X) −→ R, f 7−→∫

X| f |dm.

It is linear, subadditive, and positive homogeneous. Moreover,∫X| f |dm = 0 =⇒ f = 0 m− a.e.

(3) We say f and g is L 1(X) are equivalent if

f ∼ g f = g m− a.e.

Let

(1.4.14) L1(X) := L 1(X)/ ∼=[ f ] : f ∈ L 1(X)

.

Then

L1(X) −→ R, [ f ] 7−→∫

X| f |dm = || f ||L1(X)

is well-defined. Thus

L1(X) ∼=space of (classes of) functions defined a.e.

and integrable on X, together with thenorm f 7→ || f ||L1(X) =

∫X | f |dm.

Theorem 1.4.15. (Riesz-Fischer) If (X, A ,m) is a measure space, then L1(X) is a Ba-nach space.


Theorem 1.4.16. If (X, A ,m) is a measure space, where X is a locally compact topologicalspace, A the Borel σ-field, and m a Borel measure, then D0(X), the space of continuousfunctions with compact support, is dense in L1(X).

(4) Theorem 1.4.15 and Theorem 1.4.16 hold when f takes its values in aBanach space (E, B, || · ||).

1.4.9. Lp-spaces. Let (X, A ,m) be a measure space.

(1) Set

(1.4.15) Lp(X) =the space of (classes of) measurable functionsdefined a.e. on X, such that | f |p is integrable.

and

(1.4.16) || f ||Lp(X) :=[∫

X| f |pdm

]1/p=

[∣∣∣∣∣∣∣∣| f |p∣∣∣∣∣∣∣∣L1(X)

]1/p

.

(2) The Minkowski inequality:

(1.4.17) || f + g||Lp(X) ≤ || f ||Lp(X) + ||g||Lp(X), p ≥ 1.

The Holder inequality:

(1.4.18) || f g||L1(X) ≤ | f ||Lp(X)||g||L1(X),1p+

1q= 1, p ≥ 1.

(3) p ≥ 1 =⇒ (Lp(X), || · ||Lp(X)) is a normed space.(4) On L2(X):

(1.4.19) ( f |g)L2(X) =∫

Xf gdm, ( f | f )L2(X) = || f ||2L2(X).

(5) In the space Lp(X) strong convergence is also called convergence in themean of order p.

(6) The space Lp(X) (p ≥ 1 or p = ∞) are locally convex.(7) (Lp(X), || · ||) (p ≥ 1) are Banach spaces, and (L2(X), (·|·)L2(X)) is a Hilbert

space.

Theorem 1.4.17. (Riesz) For each p ≥ 1, the space Lp(X) is complete, that is, ∃ f ∈Lp(X) such that

limn→∞

|| fn − f ||Lp(X) = 0

if limm,n→∞ || fn − fm||Lp(X) = 0.

(8) C2([a, b]) is NOT complete.



PROOF. Consider [a, b] = [−1, 1] and

fn(x) :=

0, −1 ≤ x ≤ − 1

n ,nx+1

2 , − 1n ≤ x ≤ 1

n ,1, 1

n ≤ x ≤ 1.

Then fn(x) converges to

f (x) :=

0, −1 ≤ x < 0,1, 0 ≤ x ≤ 1,

which is not in C2([−1, 1]).

Theorem 1.4.18. (Inclusion theorem) X is of finite measure (e.g., X is a compact subsetof Rn together with the Lebesgue measure) =⇒

1 ≤ p′ < p =⇒ Lp(X) ⊂ Lp′(X).

Actually,

|| f ||Lp′ (X)≤ || f ||Lp(X)[m(X)]

1p′ −

1p .

(9) Let

L∞(X) =the space of (class of a.e. defined) measurable function bounded

almost everywhere on X which has the norm given by|| f ||L∞(X) := infM > 0 : | f (x)| ≤ M a.e. on X

(10) ∀ g ∈ Lq(X) (1 ≤ q ≤ +∞), define g′ ∈ (Lp(X))′ ( 1p + 1

q = 1) by

g′ : Lp(X) −→ C, f 7−→∫

Xf gdm.

Hence

Lq(X) ⊂ (Lp(X))′, 1 ≤ p, q ≤ +∞ and1p+

1q= 1.

Theorem 1.4.19. (Riesz representation theorem) 1 < p < +∞ and 1/p + 1/q = 1=⇒ (Lp(X))′ = Lq(X). In particular, (L2(X))′ = L2(X).

(11) If (X, A ,m) is a measure space and m is σ-finite, then

(L1(X))′ ∼= L∞(X).

(12) In general, it is not true that (L∞(X))′ is L1(X). For example, consider(X, A ,m) = (R, B, l) and x0 ∈ R. Define

L : CB(R) −→ R, f 7−→ f (x0).

Then L is linear and continuous in the L∞(R) norm,

|L( f )| = | f (x0)| ≤ || f ||L∞(R).


By the Hahn-Banach extension theorem, ∃ continuous linear functionalon L∞(R) such that the restriction to CB(R) is L. But this linear formcannot be given by

L( f ) =∫

Rf gdx

for some g ∈ L1(R). For instance, the Dirac measure cannot be repre-sented by a locally integrable function.

Exercise 1.4.20. Let (X, A ,m) be a measure space. The decreasing rearrangementof a measurable function f defined on X is the function f ∗ : [0,+∞)→ R by

f ∗(t) := infs > 0 : d f (s) ≤ t,where d f : [0,+∞)→ R denotes the distribution function of f and is given by

d f (s) := m (x ∈ X : | f (x)| > s) .

Show that(1) d f+g(α + β) ≤ d f (α) + dg(β) and d f g(αβ) ≤ d f (α) + dg(β) for any mea-

surable functions f , g and any α, β > 0.(2) For f ∈ Lp(X), 0 < p < +∞, we have

|| f ||pLp(X)= p

∫ ∞

0αp−1d f (α)dα.

(3) On (Rn, B, ln = dx) compute d f and f ∗ for

f (x) =1

1 + |x|p , 0 < p < +∞.

(4) Show that f ∗(d f (α)) ≤ α whenever α > 0 and d f ( f ∗(t)) ≤ t whenevert > 0, where f is a measurable function.

(5) ( f + g)∗(t1 + t2) ≤ f ∗(t1) + g∗(t2) and ( f g)∗(t1 + t2) ≤ f ∗(t1)g∗(t2) formeasurable functions f , g and any t1, t2 > 0.

(6) d f = d f ∗ and∫X| f |p dm =

∫ ∞

0| f ∗(t)|pdt, 0 < p < +∞,

for any measurable function f .Given 0 < p, q ≤ +∞ and a measurable function f define

|| f ||Lp,q(X) :=(∫ +∞

0

(t1/p f ∗(t)

)q dtt

)1/q, q < +∞

and|| f ||Lp,∞(X) := sup

t>0

(t1/p f ∗(t)

), q = +∞.

The Lorentz space is defined to be

Lp,q(X) :=The space of (classes of) measurable functionsdefined a.e. on X, such that || f ||Lp,q(X) < +∞

Note that Lp,p(X) = Lp(X) and L∞,∞(X) = L∞(X). Compute


(7) If f = ∑1≤j≤N ajχEj , where the sets Ej have finite measure and arepairqise disjoint and a1 > · · · > aN , then

|| f ||Lp,q(X) =

(pq

)1/q [aq

1Bq/p + aq2(Bq/p

2 − Bq/p1 ) + · · ·+ aq

N(Bq/pN − Bq/p

N−1)]1/q

when 0 < p, q < +∞, and

|| f ||Lp,∞(X) = sup1≤j≤N

(ajB

1/pj

)when q = +∞. Here Bj := ∑1≤i≤j m(Ei).

(8) For 0 < p < +∞ and 0 < q ≤ +∞ we have

|| f ||Lp,q(X) = p1/q(∫ +∞

0

[d f (s)1/ps

]q dss

)1/q.

A subset A of (X, A ,m) id called an atom if m(A) > 0 and every subset Bof A has measure either equal to zero or equal to m(A). We say that (X, A ,m) isnon-atomic if it contains no atoms. Equivalently, (X, A ,m) is non-atomic if andonly if ∀ A ∈ A with m(A) > 0 ∃ proper subset B ( A in A with m(B) > 0 andm(A \ B) > 0. (Rn, B, ln) is non-atomic. It can be shown that if (X, A ,m) is anon-atomic σ-finite measure space, then

(L1,q(X))′ = L∞(X) (0 < q ≤ 1) =⇒ (L1(X))′ = L∞(X),

and

(Lp,q(X))′ = Lp′ ,q′(X) (1 < p, q < +∞, 1/p + 1/p′ = 1 = 1/q + 1/q′).

PROOF. (1) follows from the following inclusions

x ∈ X : |( f + g)(x)| > α + β ⊂ x ∈ X : | f (x)| > α ∪ x ∈ X : |g(x)| > βand

x ∈ X : |( f g)(x)| > αβ ⊂ x ∈ X : | f (x)| > α ∪ x ∈ X : |g(x)| > β.(2) Using Fubini’s theorem we have

p∫ +∞

0αp−1d f (α)dα = p

∫ +∞

0αp−1

∫X

χx∈X:| f (x)|>αdm(x)

=∫

X

∫ | f (x)|

0pαp−1dαdm(x) =

∫X| f (x)|pdm(x) = || f ||pLp(X,m)

.

(3) Direct computation shows

d f (α) =

ωn

(1α − 1

)n/p, α < 1,

0, α ≥ 1,

andf ∗(t) =

1(t/ωn)p/n + 1

where ωn is the volume of the unit ball in Rn.(4) Since α ∈ s > 0 : d f (s) ≤ d f (α), it follows that

f ∗(d f (α)) = infs > 0 : d f (s) ≤ d f (α) ≤ α.


d f ( f ∗(t)) ≤ t directly follows from the definition of f ∗(t).(5) follows from (1).(6) d f = d f ∗ follows from the fact that f ∗(t) > s if and only if t < d f (s), i.e.,

t ≥ 0 : f ∗(t) > s = [0, d f (s)). The last identity follows from (2).(7) Indeed,

|| f ||qLp,q(X)=

∫ +∞

0

[t1/p f ∗(t)

]q dtt

=∫ +∞

0

[t1/p ∑

1≤j≤Najχ[Bj−1,Bj)

(t)

]qdtt

= ∑1≤j≤N

∫ Bj

Bj−1

(t1/paj)q dt

t= ∑

1≤j≤Naq

j1

q/ptq/p

∣∣∣∣Bj

Bj−1

=pq ∑

1≤j≤Naq

j

(Bq/p

j − Bq/pj−1

)and

|| f ||Lp,∞(X) = supt>0

[t1/p f ∗(t)

]= sup

t>0

[∑

1≤j≤Najt1/pχ[Bj−1,Bj)

(t)

]= sup

1≤j≤N

(ajB

1/pj

).

(8) q = ∞:

|| f ||Lp,∞(X) = supt>0

[t1/p f ∗(t)

]= sup

α>0

[αd f (α)

]1/p.

q < +∞: For a simple function f = ∑1≤j≤N ajχBj , from (7) we have

|| f ||Lp,q(X) =

(pq

)1/q[

∑1≤j≤N

aqj

(Bq/p

j − Bq/pj−1

)]1/q

.

On the other hand

p1/q(∫ +∞

0

[d f (s)1/ps

]q dss

)1/q

= p1/q

∫ +∞

0

s

(∑

1≤j≤NBjχ[aj+1,aj)

(s)

)1/pq

dss

1/q

= p1/q

1.5. Linear functional analysis

An operator on a vector space X is a mapping T : X → Y from a subset of X,called the domain Dom(T) of T, onto a subset of another vector space Y, calledthe range Range(T) of T.

1.5.1. Bounded linear operators. A continuous linear operator T on a vectorspace X can be extended to a continuous linear operator with domain X.

(1) Let

L(X, Y) := continuous linear operators from X to Y.

1.5. LINEAR FUNCTIONAL ANALYSIS 41

(2) A linear operator T : X → Y between normed spaces is bounded if ∃K ≥ 0 such that

||T(x)||Y ≤ K||x||X , ∀ x ∈ X.

Theorem 1.5.1. A linear operator T : X → Y between normed spaces is continuous ifand only if T is bounded.

(3) Let T : X → Y be a continuous linear operator between normed spaces.Define the norm of T to be

(1.5.1) ||T|| := infK > 0 : ||T(x)||Y ≤ K||x||X = sup||T(x)||Y : ||x||X = 1.(4) An unbounded operator: X = (C∞([0, 1]), || · ||L∞) with || f ||L∞ := sup[0,1] | f |,

T := d/dx. Then T is linear but not bounded. Since

||T(sin kt)||L∞ = k = k|| sin kt||L∞

for all k > π/2.(5) If X is a finite-dimensional normed vector space, then ∀ linear operator

is continuous and hence bounded.

Theorem 1.5.2. If X is a normed space and Y is a Banach space, then (L(X, Y), || · ||) isBanach.

(6) X normed space =⇒ X′ := L(X, K) Banach space =⇒ bidual X′′ :=L(X′, K) Banach space.

Theorem 1.5.3. If X is a normed space, then there exists an isomorphism J : X →J(X) ⊂ X′′.

PROOF. Define

J : X −→ X′′, x 7−→(⟨·, x⟩ : X′ −→ K, x′ 7−→ ⟨x′, x⟩

).

Then J is linear, injective, and is an isometry. (7) A normed space X is reflective if X′′ = X.(8) ∀ 1 < p < +∞ =⇒ Lp(X) is reflective.

A Banach algebra is a Banach space together with an associative internal op-eration (called multiplication), (B, ·) or B.

(1) X Banach space =⇒B(X) := L(X, X) is a Banach algebra with

(T1T2)x := T1(T2x) =⇒ ||T1T2|| ≤ ||T1|| · ||T2||.(2) An involutive Banach algebra is a Banach algebra with a norm preserv-

ing involution ∗: (B,+, ·, || · ||, ∗). An involution ∗ in B is a mappingB→ B given by T 7→ T∗ (adjoint of T) such that


(2.1) (T + S)∗ = T∗ + S∗,(2.2) (αT)∗ = αT∗,(2.3) (ST)∗ = T∗S∗,(2.4) T∗∗ = T,(2.5) ||T∗|| = ||T||.T ∈ B is self-adjoint (resp. normal) if T = T∗ (resp. TT∗ = T∗T).

(3) A C∗-algebra is an involutive Banach algebra which in addition satisfies(2.6) ||T∗T|| = ||T||2.

(4) Examples:(4.1) In C, the mapping z 7→ z is an involution.(4.2) X Hilbert space =⇒ canonical involution on B(X) is (T f |g) = ( f |T∗g).

Let X be a Banach space.

(1) A linear operator T on X has the transposed operator T on X′ defined by⟨Tx′, y

⟩= ⟨x′, Ty⟩, x′ ∈ X′, y ∈ X.

(2) X Hilbert space =⇒ (T)∗ = T∗.

Let X be a Banach space and T ∈ B(X).(1) λ ∈ C is said to be in the spectrum σ(T) if T− λI is not bijective.

(1.1) If T− λI is not injective (i.e., Tx− λx = 0 has a solution x = 0), λ iscalled an eigenvalue of T, and is said to be in the point spectrum ofT. The corresponding solutions are called eigenvectors. The set ofall point spectrums of T is denoted σp(T).

(1.2) If T− λI is injective but not surjective.(1.2.1) λ is said to be in the continuous spectrum of T, if Range(T−

λI) is dense in X. The set of all continuous spectrums of T isdenoted σc(T).

(1.2.2) Otherwise, λ is said to be in the residual spectrum of T. Theset of all residual spectrums of T is denoted σr(T).

Hence

(1.5.2) σ(T) = σp(T) ⊔ σc(T) ⊔ σr(T).

(2) dim(X) < +∞ =⇒ σ(T) = σp(T).

Theorem 1.5.4. X Banach space =⇒ ∀ T ∈ B(X) has the following properties:

σ(T) = ∅ and σ(T) ⊂ D||T|| ⊂ C.

Theorem 1.5.5. X Hilbert space, T ∈ B(X) self-adjoint =⇒(i) σr(T) = ∅.

(ii) σ(T) ⊂ R and ||T|| = supλ∈σ(T) |λ|.(iii) Eigenvectors corresponding to distinct eigenvalues are orthogonal.

1.5. LINEAR FUNCTIONAL ANALYSIS 43

Theorem 1.5.6. X Banach space =⇒(i) T ∈ B(X) with ||T− I|| < 1 is invertible and its inverse is

(1.5.3) T−1 = I + ∑n≥1

(I− T)n.

(ii) Breg(X) := invertible elements in B(X) is open in B(X).(iii) Breg(X)→ Breg(X), T 7→ T−1, is homeomorphic.

1.5.2. Compact operators. Let f : X → Y be a mapping between two metricspaces (X, dX) and (Y, dY).

(1) f is compact if(1.1) f is continuous, and(1.2) f maps every bounded subset of X into a relatively compact subset

in Y.(2) If X, Y are Hausdorff topological vector spaces and f : X → Y is linear

and maps every bounded subset of X into a relatively compact subset ofY, then f is compact.

(3) Let F be a family of functions defined on (X, dX). We say F is equi-continuous if ∀ ϵ > 0 ∃ δ > 0 such that

| f (x)− f (x′)| < ϵ

whenever x, x′ ∈ X with dX(x, x′) < δ and whenever f ∈ F .

Theorem 1.5.7. (Ascoli-Arzela theorem) X compact metric space, C(X) space of con-tinuous functions on X with the uniform norm =⇒∀ bounded and equi-continuous subsetK ⊂ C(X) is compact.

Let (X, dX) be a compact metric space, and (Y, dY) a complete metric space.(1) F , a family of mappings from X to Y, is equicontinuous if ∀ ϵ > 0 ∃

δ > 0 such thatdY( f (x), f (x′)) < ϵ

whenever x, x′ ∈ X with d)X(x, x′) < δ and whenever f ∈ F .(2) Theorem 1.5.7 is still valid, if “K is bounded” is replaced by “∀ x ∈ X, f (x) : f ∈ K is relatively compact in Y”.

(3) Example:(3.1) Let (X, dX) be a compact metric space, (Y, dY) be a compact metric

space with a regular Borel measure n, and K : X×Y → R be contin-uous. Define

T : C(Y) −→ C(X), f 7−→ (T f )(x) :=∫

YK(x, ·) f dn.

Then T is linear, continuous, and compact, because for any boundedsubset K of C(Y), T(K) is a bounded, equicontinuous subset of C(X)and then is compact.


(3.2) Let (X, A ,m) and (Y, B, n) be measure spaces. Define

T : L2(Y) −→ L2(X), f 7−→∫

YK(x, ·) f dn,

where K ∈ L2(X × Y). Then T is a linear compact operator. Also Tis a Hilbert-Schmidt operator, i.e., if (en)n≥1 is a base of L2(Y), then

∑n≥1||Tei||L2(X) < +∞.

Theorem 1.5.8. (Fredholm alternative) X Banach space, 0 = λ ∈ C, T ∈ B(X)linear compact operator =⇒ one and only one of the two following statements is true:

(i) T f − λ f = g has one solution f for each g ∈ X (i.e., T− λI is isomorphic);(ii) T f − λ f = 0 has no zero solution (i.e., λ is an eigenvalue of T). For each λ

except possibly λ = 0, the solutions span a finite dimensional subspace of X.

Theorem 1.5.9. (Riesz-Schauder) The spectrum of a compact operator has no accumu-lation point other than, possibly, zero.

Theorem 1.5.10. (Adjoint theorem) T f − λ f = 0 has a nonzero solution⇐⇒ T f −λ f = 0 has a nonzero solution.

Let λα be an eigenvalue of T. T f − λα f = g⇐⇒ g is such that ⟨ fα, g⟩ = 0 for allfα satisfying T fα − λα fα = 0.

Theorem 1.5.11. (Hilbert-Schmidt) Let H be a Hilbert space and T a self-adjoint com-pact operator. Then ∃ orthonormal base of H made of eigenvectors of T.

1.5.3. Open mapping and closed graph theorems. Over Banach spaces, thereare three fundamental theorems: uniform boundedness theorem, open mappingtheorem, and closed graph theorem.

Theorem 1.5.12. (Uniform boundedness theorem) Let Tn : X → Y be a sequenceof linear continuous mappings between Frechet spaces. If (||Tnx||Y)n≥1 is uniformlybounded (i.e., independent of n) for each x ∈ X, then (||Tn||)n≥1 is also uniformlybounded.

In particular, if limn→∞ Tnx exists for each x ∈ X, then the limit Tx :=limn→∞ Tnx is linear and continuous.

1.6. DIFFERENTIABLE CALCULUS ON BANACH SPACES 45

Theorem 1.5.13. (Open mapping theorem) If T : X → Y is a linear continuoussurjective mapping between Frechet spaces, then T is open.

Corollary 1.5.14. (Banach theorem) If T : X → Y is a linear continuous bijectivemapping between Frechet spaces, then T is isomorphic.

Let T : X → Y be a mapping between metric spaces.(1) T is closed if ∀ (xn)n≥1 ⊂ Dom(T) with dX(xn, x) → 0 for some x ∈ X,

and (Txn)n≥1 converges to y ∈ Y, then x ∈ Dom(T) and Tx = y.(2) The graph of (T, Dom(T)) is the subset of X × Y with elements (x, Tx),∀ x ∈ Dom(T).

(3) T is closed⇐⇒ Graph(T) is closed in X×Y.

Theorem 1.5.15. (Closed graph theorem) If T : X → Y is a closed linear map withdomain X between Frechet spaces, then T is continuous.

1.6. Differentiable calculus on Banach spaces

Let (X, || · ||X) and (Y, || · ||Y) be Banach spaces, U ⊂ X open, x0 ∈ U, andf : X → Y a mapping.

(1) f is said to be differentiable at x0 if ∃ Df|x0 ∈ L(X, Y) such that

(1.6.1) f(x0 + h)− f(x0) = Df|x0 h + R(h)

where x0, x0 + h ∈ U, f(x0), f(x0 + h), R(h) ∈ Y, and ||R(h)||Y = o(||h||X).We call

Df|x0 = Df(x0) = f′(x0) = Dx0 fthe Frechet differential at x0 or derivative at x0.

(2) The differential of f : X → K is called a functional derivative.(3) f is differentiable in U if it is differentiable at every x0 ∈ U. The differ-

ential Df is a mapping U → L(X, Y) given by x 7→ Df|x.(4) If Df is continuous, f is said to be continuously differentiable or of class

C1.Examples:(1) f : U ⊂ R→ R is differentiable =⇒ D f |x0 = d f /dx|x0 ∈ R.(2) L : X → Y linear mapping =⇒ DL|x0 = L.(3) C : X → Y constant mapping =⇒ DC|x0 = 0.(4) f : R→ X =⇒ Df|t = f′(t) ∈ L(R, X), h 7→ f′(t)h. Then

f′(t)h = h[f′(t)1], ∀ h ∈ R.

Here f′1 : t 7→ f′(t)1 ∈ X can be identified with f′ : R→ X.(5) f : Rn → Rp, x 7→ y = f(x), where

yα = f α(x1, · · · , xn), 1 ≤ α ≤ p with f α ∈ C1.


The Jacobian matrix of f at x0 is

Df|x0 h =

(∑

1≤i≤n

∂ f α

∂xi

∣∣∣∣x0

hi

)1≤α≤p

=(

∂i f α|x0 hi)

1≤α≤p.

When X, Y are non-Banach, locally convex topological vector spaces, we sayf : U → Y, U open in X, is differential at x0 ∈ U, if

f(x0 + h)− f(x0) = Df|x0 h + R(h), h ∈ X,

where x0, x0 + h ∈ U, f(x0), f(x0 + h), R(h) ∈ Y, Df|x0 ∈ L(X, Y), and R is tangentto zero.

(1) Let o(t) be a real function of t ∈ R∩ (−1, 1) such that limt→0 o(t)/t = 0.(2) R is said to be tangent to zero if ∀ neighborhood N of 0Y ∈ Y ∃ neighbor-

hood V of 0X ∈ X such that R(tV) ⊂ o(t)N.

Theorem 1.6.1. (Composite mapping) Let X, Y, Z be Banach spaces, U ⊂ X open,V ⊂ Y open, f : U → Y differential at x0 , g : V → Z differential at y0 = f(x0),x0 ∈ U, and y0 ∈ V =⇒ h := g f is differential at x0 and

(1.6.2) Dh(x0) = Dg(y0) Df(x0).

Theorem 1.6.2. (Inverse mapping) U ⊂ X open, V ⊂ Y open, f : U → V invertibleand differential at x0 ∈ U, and Df(x0) : X → Y is an isomorphism =⇒ f−1 is differentialat y0 = f(x0) ∈ V and

(1.6.3) Df−1(y0) =1

Df(x0).

PROOF. f is differential at x0 =⇒ f(x0 + h) − f(x0) = Df(x0)h + R(h). Lety0 + k = f(x0 + h). Then

(y0 + k)− y0 = Df(x0)h + R(h)

and hence[Df(x0)]

−1(k) = h + [Df(x0)]−1(R(h)).

Therefore

f−1(y0 + k)− f−1(y0) = h = [Df(x0)]−1k− [Df(x0)]

−1(R(h)).

We need to check that

lim||k||Y→0

||[Df(x0)]−1(R(h))||X||k||Y

= 0.

Because lim||h||X ||R(h)||Y/||h||X = 0, we have ||R(h)||Y ≤ c||h||X for some c > 0with 0 < 1− c||[Df(x0)]

−1|| < 1. According to∣∣∣∣∣∣[Df(x0)]−1(k)

∣∣∣∣∣∣X=∣∣∣∣∣∣h + [Df(x0)]

−1(R(h))∣∣∣∣∣∣

X≥ ||h||X−

∣∣∣∣∣∣[Df(x0)]−1(R(h))

∣∣∣∣∣∣X


≥(

1− c∣∣∣∣∣∣[Df(x0)]

−1∣∣∣∣∣∣) ||h||X

so that lim||k||Y→0 ||h||X/||k||Y = 0. From

||[Df(x0)]−1(R(h))||X||k||Y

=||[Df(x0)]

−2(R(h))||X||h||X

· ||h||X||k||Ywe obtain the desired limit.

1.6.1. Diffeomorphisms. A mapping f : U → V is said to be diffeomorphicif f is a bijection with f and f−1 continuously differentiable (of class C1).

(1) homomorphism + of class C1 ; diffeomorphism ( f (x) = x3).

Theorem 1.6.3. A homeomorphism f : U → V of class C1 is a diffeomorphism ⇐⇒Df(x) is an isomorphism for every x ∈ U.

♣ Exercise: Proof Theorem 1.6.3. (Hint: Use Theorem 1.6.2)

(2) X = ∏1≤i≤n Xi product of Banach spaces, Y Banach space, U ⊂ X open,f : U → Y a mapping, and a given point a = (a1, · · · , an).

(2.1) The partially constant mapping ei : Xi → X, where

ei(xi) := (a1, · · · , ai−1, xi, ai+1, · · · , an).

Then

ei(xi + hi)− ei(xi) = (0, · · · , 0, hi, 0, · · · , 0)

so

Dei : Xi −→ X, hi 7−→ (0, · · · , 0, hi, 0, · · · , 0) =: hi · (0, · · · , 0, 1, 0, ·, 0)

called the partially identity mapping. We can view Dei as a map-ping Xi → X given by above.

(2.2) The partial derivative of f : U → Y at a is

∂f∂xi

∣∣∣∣a≡ f ′xi (a) := D(f ei)

∣∣∣∣ai= Df(a) Dei|ai = Df(a) Dei(ai).

(2.3) Moreover

∑1≤i≤nb

f′xi (a)hi = ∑1≤i≤n

(Df(a) Dei(ai)

)hi = Df(a)h.

(2.4) The existence of f′ ⇐⇒ the existence of partial derivatives; but theconverse is not true.

(2.5) f′xi : U → L(Xi, Y), x 7→ f′xi (x) =⇒ existence and continuity of Df ifand only if existence and continuity of f′xi .

(2.6) Examples:(2.6.1) f : Rn → Rp, x = (x1, · · · , xn) 7→ f(x) = ( f α(x))1≤α≤p =⇒

f′xi (x0) = (∂ f α/∂xi(x0))1≤α≤p ∈ Rp


(2.6.2) U ⊂ Cm(Rn) open, V ⊂ C(Rn) open, and P : U → V, u 7→P(Dmu), nonlinear partial differential operator of order m, where

Dmu := derivatives of u of order ≤ m,P := corresponding function C1 in all its arguments.

Then P is differential at u0 ∈ U and

DP(u0) : Cm(Rn) −→ C(Rn), h 7−→ ∑|j|≤m

∂P∂Dju

(Dmu0)Djh,

the linearization of P that is a linear partial differential opera-tor of order m.

Theorem 1.6.4. (The mean value theorem) X, Y Banach spaces, U ⊂ X convex opensubset, f : U → V of class C1 =⇒ ∀ x ∈ U and ∀ x + h ∈ U, we have

f(x + h)− f(x) =∫ 1

0Df(x + th)hdt.

PROOF. Let γ : [0, 1]→ U be defined by γ(t) := x + th ∈ U (U is convex) andu = f γ : [0, 1]→ Y. Then Du(t) = Df(x + th)γ′(t) = Df(x + th)h ∈ Y.

Corollary 1.6.5. U topological space, Y Hausdorff topological space, f : U → Y contin-uous =⇒

(1) f is locally constant and U is connected =⇒ f is constant on U.(2) Df ≡ 0, U is a connected and locally convex topological vector space =⇒ f is

constant on U.

1.6.2. Euler’s equation. Let [a, b] ⊂ R be closed and

L : [a, b]×R×R −→ R, (x, y, z) 7−→ L(x, y, z)

be C1 on R×R.(1) Define

S : C1([a, b]) −→ R, q 7−→∫ b

aL(x, q(x), q′(x))dx

and||q||C1([a,b]) := sup

[a,b](|q|+ |q′|).

(2) Compute, ∀ h ∈ C1([a, b]),

S(q + h) =∫ b

aL(x, q(x) + h(x), q′(x) + h′(x))dx

and

S(q + h)− S(q) =∫ b

a

[h(x)L′y(x, q(x), q′(x)) + h′(x)L′z(x, q(x), q′(x))

]dx


+∫ b

aα(|h(x)|+ |h′(x)|

)dx

where α = o(||h||C1([a,b])). Therefore

DS(q)h =∫ b

a

[h(x)L′y(x, q(x), q′(x)) + h′(x)L′z(x, q(x), q′(x))

]dx.

(3) Assume that L and q are C2 =⇒∫ b

ah′(x)L′z(x, q(x), q′(x))dx

=[h(x)L′z(x, q(x), q′(x))

]ba −

∫ b

ah(x)

ddx[L′z(x, q(x), q′(x))

]dx.

(4) Define

U :=

q ∈ C2([a, b]) : q(a) = q(b) = 0⊂ C2([a, b]).

Then for S : U → R, we have, q ∈ U,

DS(q) : U −→ R, h 7−→ DS(q)h

with

DS(q)h =∫ b

aE (L)hdx

where

E (L) := L′y −d

dxL′x, L := L(x, q(x), q′(x)).

Observe that

DS(q) = 0 ⇐⇒ E (L) = 0. (Euler′sequation)

Euler’s equation for several variables and several unknown functions. Letq : Ω → Rp, x 7−→ qα(x1, · · · , xn), be of class C1, Ω ⊂ Rn be open and bounded,and Dq = (∂qα/∂xi) = (∂iqα).

(1) L : Rnp+p+n → R of class C1.(2) S : ∏p times C1(Ω) −→ R, where

S(q) :=∫

ΩL(x, q(x), Dq(x))dx.

(3) Then

DS(q)h = ∑1≤i≤n, 1≤α≤p

∫Ω

(hαL′qα +

∂hα

∂xi L′∂iqα

)dx.

(4) Assume that L and q are of class C2 =⇒ Define

U :=

q ∈ C2(Ω)× · · · × C2(Ω)︸︷︷︸p times

∣∣∣∣q|∂Ω = 0

.


Then for S : U → R, we have

DS(q)h :=∫

Ω∑

1≤α≤phαEα(L)dx,

whereEα(L) := L′qα − ∑

1≤i≤n

∂

∂xi L′∂iqα .

(5) In general we can replace U in (4) by

Uf :=

q ∈ C2(Ω)× · · · × C2(Ω)︸︷︷︸p times

∣∣∣∣q|∂Ω = f

with the same operator Eα(L). Because q, q + h ∈ Uf implies h|∂Ω = 0.

1.6.3. Higher order differentials. Let (X, || · ||X) and (Y, || · ||Y) be Banachspaces, U ⊂ X be open, and f : U → Y be a C1-mapping.

(1) If Df : U → L(X, Y) is differential, define its second differential of f atx:

(1.6.4) D2f|x ≡ D2f(x) ≡ f′′(x) : X −→ L(X, Y)

and the second variation of f:

(1.6.5) D2f ≡ f′′ ∈ L (U,L(X,L(X, Y))) .

(2) f is of class Cp on U if Dpf exists on U and is continuous.(3) A diffeomorphism f is of class Cp on U if f and f−1 are of class Cp on U.

Theorem 1.6.6. X, Y, Z Banach spaces =⇒ ∃ natural isomorphism

L(X,L(Y, Z)) −→ L(X×Y, Z)

where

L(X×Y, Z) :=

bilinear continuous mappingsg : X×Y → Z

with norm

||g|| := inf K > 0 : ||g(x, y)||Z ≤ K||x||X ||y||Y .

PROOF. Define

L(X,L(Y, Z)) −→ L(X×Y, Z), f 7−→ α(f)

withα(f) : X×Y −→ Z, (x, y) 7−→ f(x)y.

Conversely, we can define

β(g)(x) := g(x, ·) : Y −→ Z, y 7−→ g(x, y)

for any g ∈ L(X×Y, Z). It can be shown that• ||α(f)|| ≤ ||f||.• ||β(g)|| ≤ ||g||.• α−1 = β.

Therefore, L(X,L(Y, Z)) ∼= L(X×Y, Z).


(4) ∀ x0 ∈ U, we have

f′′(x0) ∈ L(X,L(X, Y)) ∼= L(X× X, Y),(h, k) 7−→ (f′′(x0)h)k =: f′′(x0)(h, k) ∈ Y

Theorem 1.6.7. f : U → Y is twice differential at x0 =⇒ ∀ x ∈ U, f′′(x0) is a bilinearsymmetric mapping.

(5) The bilinear symmetric mapping f′′(x) is called the Hessian of f at x andalso denoted Hessx(f).

(6) When Y = K, we call f ′′(x) the quadratic form of f on X.(7) Examples:

(7.1) Finite-dimensional spaces:

f : Rn −→ Rp, x 7−→ ( f α(x1, · · · , xn))1≤α≤p,

with

f ′′(x0)(h, k) = Hessx0( f )(h, k) =

(∑

1≤i,j≤n

∂2 f α

∂xi∂xj hikj

)1≤α≤p

.

(7.2) Banach-spaces:

S : C1([a, b]) −→ R, q 7−→∫ b

aL(x, q(x), q′(x))dx.

Then

S′′(q) : C1([a, b])× C1([a, b]) −→ R, (h, k) 7−→ S′′(q)(h, k)

with (L is C2 in (y, z))

S′′(q)(h, k) =∫ b

a

[hkL′′yy(x, q(x), q′(x)) + h′k′L′′zz(x, q(x), q′(x))

+ (hk′ + h′k)L′′yz(x, q(x), q′(x))]

dx.

Theorem 1.6.8. (Taylor’s expression with integral remainder) U ⊂ X open, f :U → Y of class Cn, and [x0, x0 + h] ⊂ U =⇒

(1.6.6) f(x0 + h) = f(x0) + ∑1≤k≤n−1

f(k)(x0)

k!hk + R

with

R :=1

(k− 1)!

∫ 1

0(1− t)n−1f(n)(x0 + th)hn dt

wheref(k)(x0) ∈ L(X,L(X, · · · ,L)(X, Y))︸︷︷︸

(k−1) times

andf(k)(x0)hk :=

((f(k)(x0)h)h · · ·

)h︸︷︷︸

k times

.



Corollary 1.6.9. (Lagrange and Peano remainders) U ⊂ X open.(1) f : U → Y of class Cn, [x0, x0 + h] ⊂ U, and f(n)(x0 + th) bounded for all

t ∈ [0, 1] (i.e., ||f(n)(x0 + th)hn||Y ≤ M||h||nX , ∀ t ∈ [0, 1], ∃ M > 0) =⇒

(1.6.7)

∣∣∣∣∣∣∣∣∣∣f(x0 + h)− ∑

0≤k≤n−1

f(k)(x0)

k!hk

∣∣∣∣∣∣∣∣∣∣Y

≤ M||h||nX

n!

(2) f : U → Y of class Cn−1 and f is n differential at x0 =⇒

(1.6.8)

∣∣∣∣∣∣∣∣∣∣f(x0 + h)− ∑

0≤k≤n

f(k)(x0)

k!hk

∣∣∣∣∣∣∣∣∣∣Y

= o(||h||nX).

1.7. Calculus of variations

Let U be open in a Banach space (X, || · ||X) and f : Y → R a mapping.(1) f has a relative minimum at a ∈ U if ∃ neighborhood Va ⊂ U such that

f (x) ≥ f (a) for all x ∈ Va.(2) f has a strictly relative minimum at a ∈ U if ∃ neighborhood Va ⊂ U

such that f (x) > f (a) for all a = x ∈ Va.(3) Similarly, we can define relative maximum and strictly relative maxi-

mum.

1.7.1. Necessary conditions for minima. We first prove

Theorem 1.7.1. f is differential at a and f has a relative minimum at a =⇒ f ′(a) = 0.

PROOF. ∃ neighborhood Va ⊂ U such that f (x) ≥ f (a) for all x ∈ Va. Take aball B(a, r) ⊂ Va and consider

g(t) := f (a + th), a + th ∈ B(a, r).

By Fermat’s theorem, 0 = g′(0) = f ′(a)h =⇒ f ′(a) = 0.

Theorem 1.7.2. f is twice differential at a and f has a relative minimum at a =⇒f ′′(a) ≥ 0 (i.e., f ′′(a) : X× X → R is positive, f ′′(a)(h, h) ≥ 0).

PROOF. By Taylor’s theorem,

f (a + h)− f (a) =f ′′(a)

2h2 + ϵ(h)||h||2X , ϵ(h) = o(||h||X).

1.7. CALCULUS OF VARIATIONS 53

For ||h||X ≪ 1 one has

12

f ′′(a)(h, h) + ϵ(h)||h||2X ≥ 0.

Fix h and choose λ ∈ R with |λ| ≪ 1 =⇒

12

f ′′(a)(λh, λh) + ϵ(λh)||λh||2X ≥ 0

and thenf ′′(a)(h, h) + 2ϵ(λh)||h||2X ≥ 0.

Letting λ→ 0 yields f ′′(a)(h, h) ≥ 0 for all h ∈ X.

1.7.2. Sufficient conditions for minima. U ⊂ X open, f : U → R, and a ∈ U.

(1) If f is twice differential at a, then f ′′(a) is said to be non-degenerate if

f ′′(a) : X −→ L(X, R) = X′, h 7−→ f ′′(a)h

is an isomorphism of Banach spaces.(2) f ′′(a) is non-degenerate =⇒ ( f ′′(a)h = 0 ⇐⇒ h = 0). But the converse

only holds when dim X is finite, and in this case f ′′(a) is non-degenerateid and only if det( f ′′xixj) = 0.

Theorem 1.7.3. f ′′(a) non-degenerate and positive =⇒ ∃ λ > 0 such that ∀ h ∈ X,f ′′(a)(h, h) ≥ λ||h||2X .

PROOF. f ′′ non-degenerate =⇒ X → X′, given by h 7→ f ′′(a)h, isomorphic=⇒ ∃ µ > 0 such that

||h||X ≤ µ|| f ′′(a)h||X′ = µ sup| f ′′(a)(h, k)

∣∣||k||X = 1

.

∃ k ∈ X such that ||k||X = 1 and ||h||X ≤ µ| f ′′(a)(h, k)|. f ′′(a) positive =⇒

||h||2X ≤ µ2 [ f ′′(a)(h, h)] [

f ′′(a)(k, k)]

, ||k||X = 1.

But f ′′(a) continuous =⇒ f ′′(a)(k, k) bounded on ||k||X = 1 =⇒ ∃ M > 0 suchthat ||h||2X ≤ Mµ2[ f ′′(a)(h, h)].

Theorem 1.7.4. (Sufficient conditions) U ⊂ X open and f : U → R of class C2,a ∈ U.

(1) f ′(a) = 0, f ′′(a) positive and non-degenerate =⇒ f has a strict relative mini-mum.

(2) Assume furthermore U is convex.(2.1) f ′(a) = 0 and f ′′|U ≥ 0 =⇒ f has a minimum at a.(2.2) f ′(a) = 0 and f ′′ ≥ 0 in a neighborhood of a =⇒ f has a relative mini-

mum at a.


PROOF. (1) By Taylor’s theorem,

f (a + h) = f (a) +12

f ′′(a)h2 + ϵ(h)||h||2X , ϵ(h) = o(||h||2X).

Theorem 1.7.3 =⇒ f ′′(a)(h, h) ≥ λ||h||2X with λ > 0 =⇒

f (a + h) ≥ f (a) +[

λ

2+ ϵ(h)

]||h||2X .

Taking h sufficiently small so that |ϵ(h)| < λ/4 yields

f (a + h) ≥ f (a) +λ

4||h||2X > f (a)

for all h = 0 and ||h||X ≪ 1.(2) Use Theorem 1.6.8 + convexity =⇒

f (x) = f (a) +∫ 1

0

1− t2

f ′′(a + t(x− a))(x− a)2dt ≥ f (a)

for x ∈ U of x in a neighborhood of a.

Example: Let Ω ⊂ Rn be a bounded and open subset. Define

S : C1(Ω) −→ R, q 7−→∫

Ω∑

1≤i≤n

(∂q∂xi

)2dx.

(1) S is differential and

S′ : C1(Ω) −→ L(C1(Ω), R), q 7−→ S′(q)h = ∑1≤i≤n

2∫

Ω

∂q∂xi

∂h∂xi dx.

(2) ∀ q ∈ C2(Ω), S′(q) ∈ L(C1(Ω), R) and

S′(q)h = ∑1≤i≤n

2∫

Ω

[∂

∂xi

(∂q∂xi h

)− ∂2q

(∂xi)2 h]

dx.

(3) ∀ q ∈ C2(Ω) and ∂Ω regular (so that Stokes’ formula can be applied) =⇒

S′(q)h = ∑1≤i≤n

2∫

Ω

[(∂q∂xi h

)niω− ∂2q

(∂xi)2 h]

dx

with niω = (−1)idx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn.(4) ∀ q ∈ C2(Ω) and ∂Ω regular, q takes given functions on ∂Ω =⇒ h = 0 on

∂Ω and

S′(q)h = −2∫

Ω∑

1≤i≤n

∂2q(∂xi)2 dx.

Hence

∆q0 := ∑1≤i≤n

∂2q0

(∂xi)2 = 0 (Euler′sequation) and S′′(q0)(h, h) ≥ 0,

so that S has a minimum at q0.

1.8. IMPLICIT FUNCTION THEOREM AND INVERSE FUNCTION THEOREM 55

1.8. Implicit function theorem and inverse function theorem

Theorem 1.8.1. (Classical theorem) Let f : (x, y) 7→ f (x, y) be a function such that• f (x0, y0) = 0,• f is differential at (x0, y0),• f ′y(x0, y0) = 0.

Then f (x, y) = 0 has exactly one continuous solution y = φ(x) for x in a neighborhoodof x0 such that φ(x0) = y0.

1.8.1. Contracting mapping theorems. (X, d) complete metric space, F ⊂ Xclosed. A contracting mapping is a mapping f : F → F such that

d( f (x), f (y)) ≤ kd(x, y), k ∈ [0, 1), ∀ x, y ∈ F.

We also say that f is Lipschitz of order k < 1.

Theorem 1.8.2. (Contracting mapping theorem) (1) A contracting mapping f hasstrict one fixed point (∃! a ∈ F such that f(a) = a).

(2) (X, || · ||) Banach space, f : B(a, R) → X contracting mapping of ratio k < 1(i.e., ||f(x)− f(y)|| ≤ k||x− y||, ∀ x, y ∈ B(a, R)) =⇒φ := 1− f is a homeomorphismof an open set V ⊂ B(a, R) onto B(a − b, (1− k)R) with b = f(a). Moreover φ−1 is(1− k)−1-Lipschitz.

PROOF. (1) Let x0 ∈ F and fn(x0) := f(fn−1(x0)) ∈ F =⇒

d( fn(x0), fn−1(x0)) ≤ kd(fn−1(x0), fn−2(x0)) ≤ · · · ≤ kn−1d(f(x0), x0).

k ∈ [0, 1) =⇒ (fn(x0))n≥1 is Cauchy =⇒ ∃ a ∈ F such that

a = limn→∞

fn(x0) = limn→∞

f(fn−1(x0)) = f(a).

If ∃ another b ∈ F with f(b) = b, then

d(a, b) = d(f(a), f(b)) ≤ kd(a, b) =⇒ (1− k)d(a, b) ≤ 0.

Since k < 1, it follows that a = b.(2) ∀ y ∈ B(a− b, (1− k)R) =⇒ the point x such that φ(x) = y is unique if it

exists:

||φ(x)−φ(x′)|| ≥ ||x− x′|| − ||f(x)− f(x′)|| ≥ (1− k)||x− x′||.

Letx0 := a, x1 := f(x0) + y, · · · , xn−1 := f(xn) + y, · · · .

Then• xn ∈ B(a, R) for all n ≥ 0: Assume xn ∈ B(a, R). We now prove xn+1 ∈

B(a, R).

||xn+1 − a|| = ||xn+1 − x0|| ≤ ∑0≤i≤n

||xi+1 − xi||


≤ ∑0≤i≤n

ki||y− (b− a)|| ≤ 11− k

||y− (b− a)||.

Because y ∈ B(a− b, (1− k)R) =⇒ ||xn+1 − a|| < R.• φ is surjective: The above also shows ||xn+1 − xn|| ≤ kn(1 − k)R =⇒

(xn)n≥1 is Cauchy and its limit x satisfies

x = f(x) + y =⇒ φ(x) = y.

Moreover ||x− x′|| ≤ ||φ(x)−φ(x′)||/(1− k) =⇒φ−1 is (1− k)−1-Lipschitz. 1.8.2. Inverse function theorem. We first prove

Lemma 1.8.3. (X, || · ||X), (Y, || · ||Y) Banach spaces =⇒ GL(X, Y) := Isom(X, Y) isopen in L(X, Y).

PROOF. Recall that L(X, Y) is a Banach space with norm ||f|| for any f ∈L(X, Y). Assume GL(X, Y) = ∅. Take g0 ∈ GL(X, Y) and choose g ∈ L(X, Y) sothat

||h− 1||L(X,X) < 1, h := g−10 g.

Theorem 1.8.2 =⇒ h ∈ GL(X, X) and

h−1 = ∑n≥0

kn, k := 1− h.

Hence g = g0 h ∈ GL(X, Y) and

g0 ∈ GL(X, Y) =⇒

g ∈ L(X, Y)∣∣∣∣|||g− g0|| <

1||g−1

0 ||

is open in L(X, Y).

Theorem 1.8.4. (Inverse function theorem) (X, || · ||X) and (Y, || · ||Y) Banachspaces, U ⊂ X open, V ⊂ Y open, f : U → V of class C1, a ∈ U, b := f(a) ∈ V,f′(a) : X → Y isomorphism =⇒ ∃ open neighborhoods Ua ⊂ U and Vb ⊂ V such thatf−1 is a C1-diffeomorphism of Vb onto Ua.

PROOF. Let φ := 1 − [f′(a)]−1 f : U → X. Then φ is C1 in U and φ′(a) =1− [f′(a)]−1f′(a) = 1− 1 = 0. Hence ∃ k < 1 such that for R≪ 1

||φ′(a + tR)|| < k < 1, ∀ 0 < t < 1.

φ is a contracting mapping in the ball B(a, R) =⇒ By Theorem 1.8.2, [f′(a)]−1 f =1 −φ is a homeomorphism of U′ ⊂ B(a, R) onto B([f′(a)]−1b, (1 − k)R). Since[f′(a)]−1 is an isomorphism of X onto Y, f is an homeomorphism of U′ onto V′ :=f′(a)B([f′(a)]−1b, (1− k)R). In particular, f is invertible on V′, the equation f(x) =y has strictly one solution x ∈ U′ for y ∈ V′.

Theorem 1.6.2 =⇒ f−1 is differential at b and Df−1(b) = [Df(a)]−1. By Lemma1.8.3, ∃ open neighborhood Ua ⊂ U′ ⊂ U such that f′(x) is an isomorphism for ∀x ∈ Ua. Then by Theorem 1.6.2 again, f−1 is C1-differential on Vb := f(Ua).

1.8. IMPLICIT FUNCTION THEOREM AND INVERSE FUNCTION THEOREM 57

1.8.3. Implicit function theorem. We prove

Theorem 1.8.5. (Implicit function theorem) (X, || · ||X), (Y, || · ||Y), (Z, || · ||Z) Ba-nach spaces, U ⊂ X × Y open, f : U → Z of class C1, f(a, b) = 0, f′y(a, b) : Y → Zisomorphism =⇒ ∃ open sets W ⊂ X with a ∈ W and V ⊂ U with (a, b) ∈ V, and aC1-mapping g : W → Y such that

(x, y) ∈ V and f(x, y) = 0 ⇐⇒ y = g(x), x ∈W.

PROOF. SetF : U −→ X× Z, (x, y) 7−→ (x, f(x, y)).

Then

F′ =[

1 0f′x f′y

]and F′(a, b) : X×Y −→ X× Z isomorphism.

Theorem 1.8.4 applied to F(x, y) = (x, f(x, y)) = (x, 0), ∃ open neighborhoods V ⊂U with (a, b) ∈ V and W ⊂ X with a ∈W such that F−1 is a C1-diffeomorphism ofW × 0 onto V =⇒ ∃ g : W → Y that is C1 and y = g(x), ∀ x ∈W.

1.8.4. Global theorems. f : X → Y of class C1 and f′(x) is an isomorphismfor ∀ x ∈ X ; f is a diffeomorphism, even if X = Rn.

Example: Consider

f : R2 −→ R2, (x, y) 7−→ (ex cos y, ex sin y) .

Then

f ′(x, y) : R2 −→ R2, f ′(x, y) ∼[

ex cos y −ex sin yex sin y ex cos y

]= ex

[cos y − sin ysin y cos y

].

But f is NOT injective since f (x, y) = f (x, y + 2kπ).

Theorem 1.8.6. (X, || · ||X) and (Y, || · ||Y) Banach spaces, U ⊂ X open, f : U → Y ofclass C1, satisfying

(i) f is injective,(ii) f′(x) : X → Y is isomorphic, ∀ x ∈ U.

Then f is a diffeomorphism from U into f(U) ⊂ Y.

PROOF. By (i), f is a bijection of U onto f(U) ⊂ Y.(1) f(U) is open in Y. Let b ∈ f(U) and a ∈ U be such that f(a) = b. Theorem

1.8.4 =⇒ ∃ open neighborhoods Ua of a in U and Vb of b in Y such that f : Ua → Vbis a diffeomorphism and Vb := f(Ua) ⊂ f(U) ⊂ Y.

(2) Similarly the image by f of any open set in U is open =⇒ f−1 : f(U) → Uis continuous.

(3) f is a C1-homeomorphism of U onto f(U). By Theorem 1.6.2, f is an diffeo-morphism from U into f(U).


Figure (Theorem 1.8.7): The segment from b to y0 in Y

Theorem 1.8.7. (X, || · ||X) and (Y, || · ||Y) Banach spaces, f : X → Y of class C1,satisfying

(i) ∀ x ∈ X, f′(x) is an isomorphism,(ii) ∃ M > 0 such that ||[f′(x)]−1|| ≤ M, ∀ x ∈ X.

Then f is a diffeomorphism of X onto Y.

PROOF. According to Theorem 1.8.6 we need only to prove that f is a bijection.(1) f is surjective. Let a ∈ X and b = f(a) ∈ Y. Given y0 ∈ Y. To show the

existence of x0 such that f(x0) = y, we consider the segment from b to y0 in Y:

y(t) := b(1− t) + ty0, 0 ≤ t ≤ 1.

Theorem 1.8.4 =⇒ ∃ open neighborhoods Ua of a in X and Vb of b in Y such thatf : Ua → Vb is a diffeomorphism. Set

x(t) := f−1(y(t)) for y(t) ∈ Vb.

Claim 1: ∃maximum value t0 ∈ [0, 1] such that x(t) is defined for ∀ t ∈ [0, t0].Assume that x(t) is defined on [0, t0), then

x′(t) = [f−1(x(t))]−1y′(t), ∀ 0 ≤ t < t0 ≤ 1.

So||x′(t)||X ≤ M||y′(t)||Y = M||y0 − b||Y, 0 ≤ t < t0 ≤ 1.

By the mean-value theorem, Theorem 1.6.4,

||x(t)− x(τ)||X ≤ M||y0 − b||Y|y− τ|, ∀ 0 ≤ t, τ < t0 ≤ 1.

Hence limt→t0 x(t) = x(t0) exists and f(x(t0)) = y(t0).Claim 2: t0 = 1.Otherwise t0 ∈ (0, 1). Theorem 1.8.4 applied to the neighborhoods of x(t0)

and y(t0) makes it possible to determine x(t) = f−1(y(t)) in a neighborhood ofx(t0) and y(t0), hence for values t > t0.

From Claim 1 and Claim 2 =⇒ x(1) = f−1(y(1)) =⇒ y0 = f(x(1)) = f(x0).

1.9. DIFFERENTIABLE EQUATIONS 59

Figure (Theorem 1.8.7): γYi ∼ γY

(2) f is injective. Assume that ∃ x1 = x2 such that f(x1) = f(x2) = y. γYi ∼ γY

consisting only of y =⇒ contradicting the local inverse function theorem.

1.9. Differentiable equations

Let (X, || · ||) be a real Banach space, φ : R → X, and φ′(t) is identified withφ′ : R→ X.

1.9.1. First order differential equation. Consider an open subset U ⊂ R× Xand

(1.9.1)dxdt

= f(t, x), x ∈ X, f : U → X continuous.

A solution of this differential equation is a C1-function φ : R ⊃ I → X such that

(1.9.2) (t,φ(t)) ∈ U (∀ t ∈ I) and φ′(t) = f(t,φ(t)) (∀ t ∈ I).

Here I ⊂ R is an open interval.

(1) Example: X = Rn, x = (x1, · · · , xn), f = ( f 1, · · · , f n) =⇒

dxdt

= f (t, x) =⇒ dxi

dt= f i(t, x1, · · · , xn) (1 ≤ i ≤ n).

(2) An equation of order n on X is

(1.9.3)dnxdtn = f

(t, x,

dxdt

, · · · ,dn−1xdtn−1

),

that is equivalent to n equations of first order on X,

dxdt

= x1,dx1

dt= x2, · · · ,

dxn−1

dt= f(t, x, x1, · · · , xn−1),

where (x, x1, · · · , xn−1) ∈ X× · · · × X (n times).

1.9.2. Existence and uniqueness theorems for the Lipschitz case. Let (X, || ·||X) and (Y, || · ||Y) be Banach spaces, U ⊂ X be open. We say f : X → Y isk-Lipschitz in U if

||f(x1)− f(x2)||Y ≤ k||x1 − x2||X , ∀ x1, x2 ∈ U.


Let I ⊂ R be open, U ⊂ X be an open subset in a Banach space (X, || · ||).We say f : I ×U → X is locally Lipschitz if ∀ (t0, x0) ∈ I ×U ∃ neighborhoodN ⊂ I ×U of (t0, x0) and ∃ k > 0 such that

||f(t, x1)− f(t, x2)|| ≤ k||x1 − x2||, ∀ (t, x1), (t, x2) ∈ N.

Equivalently, f is locally Lipschitz if and only if f(t, ·) is k-Lipschitz locally.

Theorem 1.9.1. (Local existence) (X, || · ||) Banach space, U ⊂ X open, I ⊂ R open,f : I ×U → X continuous and locally Lipschitz, (t0, x0) ∈ I ×U =⇒ ∃ a > 0 such that

dxdt

= f(t, x)

has a solution φ : [t0 − a, t0 + a]→ X with φ(t0) = x0.

PROOF. Take a closed ball B0 of center x0 with radius ϵ, a closed interval I0 ofcenter t0, such that (t0, x0) ∈ I0 × B0 ⊂ N ≡ N(t0, x0), the neighborhood in thedefinition of locally Lipschitz mapping. Set

sup(t,x)∈I0×B0

|f(t, x)| ≤ M ∈ [0,+∞).

∀ (t, x) ∈ N we have

||f(t, x)|| ≤ ||f(t, x0)||+ k||x− x0||.Define

xn(t) := x0 +∫ t

t0

f(τ, xn−1(τ))dτ, n = 1, 2, · · · , t ∈ I0.

For n = 1,

||x1(t)− x0|| ≤ |t− t0| supt0≤τ≤t

||f(τ, x0)|| ≤ M|t− t0|.

So ∃ a > 0 such that x1(t) ∈ B0 for ∀ t ∈ [t0 − a, t0 + a] (i.e., a < ϵ/M). In general,if xn−1(t) ∈ B0 (t ∈ [t0 − a, t0 + a])

||xn(t)− x0|| ≤ |t− t0| supt0≤τ≤t

||f(τ, xn−1(τ))|| ≤ M|t− t0| < ϵ

for ∀ t ∈ [t0 − a, t0 + a]. Moreover

||xn(t)− xn−1(t)|| ≤∫ t

t0

||f(τ, xn−1(τ))− f(τ, xn−2(τ))||dτ

≤ k∫ t

t0

||xn−1(τ)− xn−2(τ)||dτ

so that

||xn(t)− xn−1(t)|| ≤Mkn−1

n!|t− t0|n, t ∈ [t0 − a, t0 + a]

so (xn(t))n≥1 is a Cauchy sequence in B0 for ∀ t ∈ [t0− a, t0 + a] =⇒ limn→∞ xn(t) =:x(t) ∈ B and

x(t) =∫ t

0f(τ, x(τ))dτ + x0, x(0) = x0.

Thus dx(t)/dx = f(t, x(t)).

1.9. DIFFERENTIABLE EQUATIONS 61

If f is linear and continuous in x, on X, then it is globally Lipschitz on X:

||f(t, x1)− f(t, x2)|| ≤ k||x1 − x1||, ∀ x1, x2 ∈ X.

In this case, the solution of dx/dt = f(t, x) exists globally and uniquely.

Theorem 1.9.2. (Global uniqueness theorem) Under the same hypotheses as for The-orem 1.9.1 =⇒ ∃ maximum interval J ⊂ I of t0 for which ∃! solution ψ : J → X of theequation

dxdt

= f(t, x), ψ(t0) = x0.

It is called the maximal solution for the initial value (t0, x0).

PROOF. (1) Local uniqueness. Let φ1(t) and φ2(t) be two solutions with φ1(t0) =φ2(t0) = x0. If (t, φ1(t)) and (t, φ2(t)) are in N = N(t0, x0), then

||φ1(t)− φ2(t)|| ≤ k|t− t0| supt0≤τ≤t

||φ1(τ)− φ2(τ)||.

When |t− t0| < 1/k, we get φ1(t) = φ2(t).(2) Global uniqueness. By (1) we see that

φ1 and φ2 are solutions inj ∋ t0 and φ1(t0) = φ2(t0)

=⇒ φ1(t) = φ2(t), ∀ t ∈ j.

Consider the set

Σ := (j, φ) : φ is a solution on j ∋ t0 such that φ(t0) = x0 .

∀ (j1, φ1) and (j2, φ2) in Σ, we must have φ1 = φ0 on j1 ∩ j2. Define (J, ψ) :=(∪j∈Σ j,∪φ∈Σ φ).

Example 1.9.3. (1) ∃ solution which does NOT exist in R but f is continuous andlocally Lipschitz on R× X. Consider

dxdt

= x2, X = R, (x0, t0) with x0 > 0 and t0 = 0.

The maximum solution is

φ(t) =x0

1− tx0, t ∈ (−∞, 1/x0).

(2) A differential equation depending on a parameter λ ∈ T ⊂ Y (a topologicalspace):

dxdt

= f(t, x; λ).

If||f(t, x; λ)|| ≤ M, ∀ (t, x, λ) ∈ I × B× T

and

||f(t, x1; λ)− f(t, x2; λ)|| ≤ k||x1 − x2||, ∀ t ∈ I, λ ∈ T, x1, x2 ∈ B,

then φ(t, λ) is a continuous solution of λ ∈ T and t ∈ [t0 − a, t0 + a].


1.10. Problems and references

Problem 1: Clifford algebra and Spin(4). Let V s,n−s, s ∈ Z≥1 and s ≤ n,be an n-dimensional vector space over R with inner product (v|w) and basis(ei)1≤i≤n such that

(ei|ej) = 0, i = j,

(ei|ej) = 1, i = j = 1, · · · , s,

(ei|ej) = −1, i = j = s + 1, · · · , n.

Introduce a product v ·w of vectors in V s,n−s which is associate and distributivewith respect to addition and which satisfies the condition

v ·w + w · v = 2(v|w).

The resulting algebra of all possible sums and products is called the Clifford alge-bra C(V s,n−s) of V s,n−s. Observe that

ei · ej + ej · ei = ±2δij,

(ei)2 := ei · ei = ±1,

v2 = (v|v),ei · ej = −ej · ei, i = j.

The Clifford algebra is itself a linear space of dimension ∑0≤p≤n (np) = 2n with

basis1, eI , eI1 · eI2 =: eI1 I2 , · · · , e1 · e2 · · · · · en =: e1···n,

where capital letters label ordered natural numbers: Ij < Ij+1.

(1.1) Show that C(V0,1) is C and that C(V0,2) is the algebra of quaternions.(1.2) The linear subspace of C(V s,n−s) spanned by the (n

p) products (eI1···Ip) isdenoted by Cp(V s,n−s). The linear subspaces

C+(V s,n−s) :=⊕

p evenCp(V s,n−s), C−(V s,n−s) :=

⊕p odd

Cp(V s,n−s)

are called the even and odd subspaces of C(V s,n−s). C+(V s,n−s) is also asubalgebra of V(V s,n−s). The dimension of both C+(V s,n−s) and C−(V s,n−s)is 2n−1. The algebra C+(V s,n−s) is isomorphic to the Clifford algebraC(V s,n−1−s) for certain values of s.

Show that the even subalgebra of the Dirac algebra C(V1,3) is thePauli algebra C(V3,0) and continue the sequence until R are reached.

(1.3) Show that the center Z(V s,n−s) of the algebra C(V s,n−s) is C0(V s,n−s)when n is even and C0(V s,n−s) + Cn(V s,n−s) when n is odd.

Problem 2: Consider the mapping

I : X −→ R, x 7−→∫ 1

0

[1 + |x′(t)|2

]1/4dt.

HereX :=

x ∈ C1([0, 1]) : x(0) = 0, x(1) = 1

.

1.10. PROBLEMS AND REFERENCES 63

It is clear that I(x) > 1 for every x ∈ X.(2.1) Under the norm || · ||C1([0,1]), X is also a Banach space and I is continuous.(2.2) Show that I has no upper bound.(2.3) Show that limr→1− I(xr) = 1, where

xr(t) :=

0, t ∈ [0, r],

−1 +[1 + 3(x−r)2

(1−r)2

]1/2, t ∈ [r, 1].

with r ∈ (0, 1). This means that infX I = 1.

Problem 3: Let Ω ⊂ Rn be open, U ⊂ C2(Ω), and V ⊂ C0(Ω). A quasi-linearsecond order operator is

P : U −→ V, u 7−→ Pu := aij(x, u, Du)∂2u

∂xi∂xj + bi(x, u, Du)∂u∂xi .

Assume that aij and bi are C1 functions on Ω× I× I1× · · · × In, where I, I1, · · · , Inare closed intervals in R such that if u ∈ U then u(x) ∈ I, ∂u/∂xi(x) ∈ Ii for allx ∈ Ω.

(3.1) Show that if u0 ∈ U then the mapping P is differential at u0, and computeits differential P′(u0) at u0.

(3.2) If moreover aij and bi are C1,α × C2 × · · · × C2 functions on Ω× I × I1 ×· · ·× In. P′(u0) is still the differential of u 7→ Pu, considered as a mappingfrom C2,α(Ω) into C0,α(Ω).

References:[1] Choquet-Bruhat, Yvonne; DeWitt-Morette, C.; Dillard-Bleick, M. Analysis,

manifolds and physics, Part I: Basics, Revised Edition, Elsevier, 2010.

[2] Kothe, G. Topological vector spaces, Second printing, Revised, Grundlehrender mathematischen Wissenschaften, 159, Springer-Verlag, 1983.

[3] Morrey, Charles, B. Multiple integrals in the calculus of variations, Reprint ofthe 1966 Edition, Classics in Mathematics, Springer, 2008.

CHAPTER 2

Differentiable manifolds

2.1. Differentiable manifolds

2.1.1. Definitions. A (topological) manifold is a Hausdorff topological spaceX such that ∀ x ∈ X has a neighborhood homeomorphic to Rn. The dimensionof X is n.

(1) A chart (U ,φ) of a manifold X is an open set X of X , called the domainof the chart, together with a homeomorphism φ : U → U := φ(U) ⊂ Rn

(open).(2) Given an arbitrary chart (U ,φ) of X . ∀ x ∈ U ⊂ X , we have φ(x) ∈ Rn.

The coordinates (x1, · · · , xn) of φ(x) are called the coordinates of x in thechart (U ,φ). A chart (U ,φ) is also called a local coordinate system.

(3) An atlas of class Ck on a manifold X is a set ((Uα,φα))α∈I of charts of Xsuch that X = ∪α∈IUα and (φα)α∈I satisfies the compatibility condition:

φβ φ−1α : φα(Uα ∩ Uβ) −→ φβ(Uα ∩ Uβ), (x1, · · · , xn) 7−→ (y1, · · · , yn),

with yj = yj(x1, · · · , xn) being of Ck.(3.1) Two Ck-atlases ((Uα,φα))α∈I and ((Uα′ ,φα′))α′∈I′ are equivalent (use

the symbol ∼) if φα′ φ−1α : φα(Uα ∩ Uα′) → φα′(Uα ∩ Uα′) are also

Ck.(3.2) A Ck-structure on X is an equivalence class of Ck-atlas.

♣ Exercise: Show that ∼ is indeed an equivalence relation.

(4) A Ck-manifold X is a topological manifold X together with a Ck-structureon X. When k = ∞, we say C∞-manifold as smooth manifolds.

(5) A real analytic/ Cω manifold is defined similarly, except that we nowrequire that the mappings φβ φ−1

α have to be real analytic.(6) A complex analytic/complex manifold is defined similarly with Cn re-

placing Rn and the mappings φβ φ−1α being real analytic or holomor-

phic.

♣ Exercise: (i) The double cone (x, y, z) ∈ R3 : x2 − y2 − z2 = 0 is NOTa manifold under the topology included by the usual topology on R3. However(x, y, z) ∈ R3 : x2 − y2 − z2 = 0 and x ≥ 0 is a C0-manifold that is homeomor-phic to R2.

65

66 2. DIFFERENTIABLE MANIFOLDS

Figure 2.1: f is differentiable at x

♣ Exercise: (ii) We require Hausdorff condition in the definition of mani-folds. The following topological space

X :=((−∞, 01]× 1

)∪((−∞, 02]×−1

)∪((0,+∞)× 0

), 01 = 02 = 0,

is NOT Hausdorff. Here the topology on X is generated by a basis of open neigh-borhoods of the form (−ai, 0i] ∪ (0, b) (i = 1, 2).

We have NOT included here in the definition of a differentiable manifold anaxiom of countability of the domains of the charts.

Let X be a Ck-manifold, with k ∈ 0, 1, · · · , ∞, ω.(1) Let (U ,φ) be a chart at x (i.e., x ∈ U ). Then (see Figure 2.1), any function

f : X → R gives a map f φ−1 : Rn ⊃ φ(U )→ R.(2) f is differentiable at x if f φ−1 is differentiable at φ(x).

(2.1) The definition does NOT depend on the chart: f φ−1 = ( f φ−1) (φ φ−1) ∈ C0 Ck = Ck.

(2.2) f is Cr-differentiable at k if f φ−1 is Cr-differentiable at φ(x). Herek ≥ r.

(2.3) f : X → R is Cr on X if it is Cr at ∀ x ∈ X . Let

(2.1.1) Cr(X ) := all Cr-functions f : X → R .

2.1. DIFFERENTIABLE MANIFOLDS 67

Let X be a Ck-manifold.(1) The i-th coordinate function ai on Rn:

(2.1.2) ai : Rn −→ R, (u1, · · · , un) 7−→ ui.

(2) ∀ chart (U ,φ) of X let

(2.1.3) φi := ai φ : U −→ R, x 7−→ ai(φ(x)) = xi.

Then

(2.1.4) ai := φi φ−1 : φ(U ) −→ R, (x1, · · · , xn) 7−→ xi.

Direct product manifolds:

(X , TX , ((Uα,φα))α∈I)×(Y , TY , ((Uβ,φβ))β∈J

):=

(X ×Y , TX×Y , ((Uα ×Uβ,φα ×φβ))(α,β)∈I×J

),

whereφα ×φβ(x, y) := (φα(x),φβ(y)), ∀ x ∈ Uα, y ∈ Uβ.

Then

(2.1.5) dim(X ×Y) = dimX + dimY .

2.1.2. Diffeomorphisms. Let X and Y be Ck-manifolds with dim X = m anddimY = n. Consider a mapping f : X → Y .

(1) (U ,φ) chart of X , (V , ψ) chart of Y =⇒

(2.1.6) ψ f φ−1 : φ(U ) −→ ψ(V).(2) f is Cr-differentiable at k (k ≥ r) if ψ f φ−1 is Cr-differentiable at φ(x):

ψ f φ−1 : (x1, · · · , xm) ∋ φ(U ) −→ (y1, · · · , yn) ∈ ψ(V), yα = f α(x1, · · · , xm).

(3) f is a Cr-mapping if f is Cr at x ∈ X .(4) f is a Cr-diffeomorphism if f is a bijection and f, f−1 are Cr. In particular,

f is a diffeomorphism if f is a C1-diffeomorphism.(5) For g : Y → R, define the pull back under f of g by

(2.1.7) f∗g := g f.

(6) Let

C∞M := C∞-manifolds, VectR := vector spaces over R

andHomC∞

M(X ,Y) := C∞-mappings X → Y.

Define

C : C∞M −→ VectR,

X 7−→ C∞(X ),f : X → Y 7−→ f∗ : C∞(Y)→ C∞(X ).


Figure 2.2: Cr-differentiable mapping.

♣ Exercise: Show that C is a contravariant functor.

2.1.3. Lie groups. A Lie group G is a group that is also a smooth manifoldsuch that the smooth structure is compatible with the group structure,

(2.1.8) G × G −→ G, (x, y) 7−→ xy−1,

is a C∞-mapping.

(1) Two Lie groups G1 and G2 are isomorphic if ∃ diffeomorphism f betweenG1 and G2 which preserves the group structures (so is a group homomor-phism).

(2) G Lie group =⇒ G → G, x 7→ x−1, is C∞.(3) (Rn,+) is a Lie group.(4) GL(n, R) is a Lie group.(5) A one-dimensional Lie group is usually called a one-parameter group.

Theorem 2.1.1. A connected one-parameter group G is isomorphic to R or T := R/Z.

(6) A connected topological group which possesses a neighborhood of theorigin homeomorphic to R is isomorphic to R or T.

2.2. VECTOR FIELDS AND TENSOR FIELDS 69

(7) A local Lie group is a neighborhood of the origin of a Lie group. A localone parameter group is isomorphic to an interval I ⊂ R containing theorigin.

2.2. Vector fields and tensor fields

2.2.1. Tangent vectors. Let X be a smooth manifold of dimension n. The tan-gent vector space at x ∈ X is a vector space TxX that is isomorphic to Rn.

(1) Definition A. Define

(2.2.1) C∞(x) := functions defined and C∞ on some neighborhood of x .

Any element of C∞(x) is a pair ( f ,U f ) with U f ⊂⊂ X being a neighbor-hood of x.

(1.1) A tangent vector vx : C∞(x) → R is a linear mapping satisfying theLeibniz rule

vx(α f + βg) = αvx( f ) + βvx(g), α, β ∈ R,vx( f g) = f (x)vx(g) + g(x)vx( f ), f , g ∈ C∞(x).

Consequently,

vx(1) = vx(1 · 1) = 2vx(1) =⇒ vx(1) = 0

and

vx(α) = vx(α · 1 + 0 · 1) = αvx(1) = 0, ∀ α ∈ R.

(1.2) vx is also called a derivation and vx( f ) is the directional derivativeof f along vx.

(1.3) Set

(2.2.2) TxX := tangent vectors vx to X at x.

Under the operation

(αvx + βux)( f ) := αvx( f ) + βux( f ),

TxX is a vector space.(1.4) We say that f , g ∈ C∞(x) have the same germ at x, written as f ∼ g,

if ∃ W ⊂ U f ∩ Ug such that f ≡ g onW . Set

(2.2.3) [ f ] := g ∈ C∞(x) : g ∼ f

the germ of f , and

(2.2.4) Fx := [ f ] : f ∈ C∞(x) = C∞(x)/ ∼,

the germs at x. Define + and · to be

[ f ] + [g] ≡ [( f ,U f )] + [(g,Ug)] := [( f + g,U f ∩ Ug)],

[ f ] · [g] ≡ [( f ,U f )] · [(g,Ug)] := [( f · g,U f ∩ Ug)].

(1.5) A tangent vector then is a derivation on Fx.(1.6) In the chart (U ,φ), the components of vx ∈ TxX are

(2.2.5) vi := vx(φi) with φi = ai φ.


(1.7) For f ∈ C∞(x0) and a chart (U ,φ) of x0, applying the mean valuetheorem to f φ−1, we obtain

f (x) = f (x0) + [φi(x)− φi(x0)]∂( f φ−1)

∂xi

∣∣∣∣φ(x0)+[φ(x)−φ(x0)]s

for some s ∈ (0, 1). Then ∀ vx0 ∈ Tx0X

(2.2.6) vx0( f ) = vi ∂( f φ−1)

∂xi

∣∣∣∣φ(x0)

, vi := vx0(φi).

(1.8) Set

(2.2.7)∂

∂xi

∣∣∣∣x0

f :=∂( f φ−1)

∂xi

∣∣∣∣φ(x0)

.

Then ∀ vx0 ∈ Tx0X

(2.2.8) vx0 = vi ∂

∂xi

∣∣∣∣x0

and (∂/∂xi|x0)1≤i≤n forms a natural basis for Tx0X . Consequently

(2.2.9) dim Tx0X = dimX = n

and

(2.2.10) Tx0X −→ Rn, vx0 = vi ∂

∂xi

∣∣∣∣x0

7−→ (v1, · · · , vn) =: v.

(1.9) If f has a critical point at x0 (i.e., ∂( f φ−1)/∂xi|φ(x0)= 0), then

vx0( f ) = 0, and conversely,

vx0( f ) = 0 (∀ vx0) ⇐⇒ ∂( f φ−1)

∂xi

∣∣∣∣φ(x0)

= 0.

(2) Definition B. Let (U ,φ) and (U ,φ′) be two charts (see Figure 2.3). Write

f := f φ−1, f ′ := f ′ φ′−1, f = f ′ φ′ φ−1.

Then

∂ f (x1, · · · , xn)

∂xi =∂ f ′(x′1, · · · , x′n)

∂x′ j∂(aj φ′ φ−1)(x1, · · · , xn)

∂xi

or∂ f∂xi =

∂ f ′

∂x′ j∂x′ j

∂xi .

This gives for ∀ vx ∈ TxX

vx( f ) = vi ∂ f∂xi

∣∣∣∣φ(x)

= vi ∂ f ′

∂x′ j

∣∣∣∣φ′(x)

∂x′ j

∂xi

∣∣∣∣φ(x)

and then

v′ j = vi ∂x′ j

∂xi

∣∣∣∣φ(x)

for each j = 1, · · · , n.


(2.1) The set (vi)1≤i≤n determines a vector v ∈ Rn and the set (v′ i)1≤i≤ndetermines another vector v′ ∈ Rn such that

(2.2.11) v′ = D(φ′ φ−1)

∣∣∣∣φ(x)

v.

(2.2) A tangent vector vx is an equivalence class [(U ,φ, v)] with (U ,φ)being a chart and v ∈ Rn, where (U ,φ, v) is equivalent to (U ,φ′, v′)if and only if (2.2.11) is true.

(3) Definition C. The last equivalent definition comes from differentiable curves.(3.1) A (parametrized) curve γ on X is a mapping from I ⊂ R into X by

I −→ X , t 7−→ γ(t).

(3.2) A differentiable curve γ at x0 is a differentiable mapping from I ⊂R into X such that 0 ∈ I and γ(0) = x0. Such differentiable curvesforms a set C1

x0(0).

(3.3) ∀ γ ∈ C1x0(0) and ∀ f ∈ C∞(x0) =⇒ f γ ∈ C1(0) and f γ(0) =

f (x0). The tangent vector to γ at x0 is a mapping1

(2.2.12) vγx0 : Fx0 −→ R, [ f ] 7−→ d

dt

∣∣∣∣t=0

( f γ)(0) =: vγx0( f ).

(3.4) γ1, γ2 ∈ C1x0(0) are said to be tangent at x0 if

vγ1x0 ( f ) = vγ2

x0 ( f ), ∀ f ∈ Fx0 .

(3.5) LetTx0X :=

[γ] : γ ∈ C1

x0(0)

.

(3.6) Given a tangent vector vx0 ∃ γ ∈ C1x0(0) (by Theorem 1.9.1) such that

vγx0 = vx0 .

(3.7) If vx0 = vγx0 , then (see Figure 2.4)

(vγx0)

i = vγx0(φi) =

ddt

∣∣∣∣t=0

(φi γ)(t) =dγi(t)

dt

∣∣∣∣t=0

and

(2.2.13) vγx0( f ) =

ddt

∣∣∣∣t=0

( f φ−1 φ γ) =∂( f φ−1)

∂xi

∣∣∣∣φ(x0)

dγi

dt

∣∣∣∣t=0

.

♣ Exercise: Show that (Fx,+, ·) is an algebra.

Let f : X → Y be differential at x0, dimX = n and dimY = p.(1) Define a linear mapping

(2.2.14) Txf ≡ f∗,x ≡ dxf ≡ f′(x) : TxX −→ Tf(x)Y , v 7−→ w

by

(2.2.15) w(h) := v(h f), ∀ h ∈ C∞(y)

where y := f(x).

1Since vγx0 ([ f ]) is independent of the choice of the representative f we can define vγ

x0 ( f ).


Figure 2.3: Definition B

Figure 2.4: Definition C


(2) Let

C∞M,∗ = category of

objects : pointed smooth manifold (X , x),

morphisms :HomC∞

M,∗((X , x), (Y , y)) =

f : X → Y differential and f(x) = yDefine a covariant functor

T∗ : C∞M,∗ −→ VectR

(X , x) 7−→ TxX ,f : (X , x)→ (Y , y) 7−→ T∗f : TxX −→ TyY .

♣ Exercise: Show that C∞M,∗ is a category and T∗ is a covariant functor.

Proposition 2.2.1. Let f : X → Y be differential at x and choose local coordinates(U ,φ, xi) on X and (V , ψ, yα) on Y =⇒ If v ∈ Rn and w ∈ Rp represent respectivelyv ∈ TxX and w ∈ TyY such that w = Txf(v), then

w = f′(x1, · · · , xn)v (⇐⇒ w = f′(x)v),

where f := ψ f φ−1 : φ(U )→ ψ(V).

PROOF. Compute, where yα := fα(x1, · · · , xn),

w(h) = (Txf(v)) (h) = v(h f) = vi ∂(h f φ−1)

∂xi

= vi ∂(h ψ−1 f)∂xi = vi ∂(h ψ−1)

∂yα

∂fα

∂xi = wα ∂(h ψ−1)

∂yα.

Thus wα = vi(∂fα/∂xi).

Examples:(1) Given a curve γ : R → X , a vector vγ

x tangent to γ at x := γ(0), anda mapping f : X → Y =⇒ Txf(vγ

x ) is tangent to f γ at y := f(x) =f γ(0).

PROOF. ∀ g ∈ C∞(y) =⇒(Txf(vγ

x ))(g) = vγ

x (g f) =ddt

∣∣∣∣t=0

(g f γ)(t) =ddt

∣∣∣∣t=0

(g (f γ))(t).

(2) Under hypothesis in (1) =⇒

T∗f(vγx ) =

ddt

∣∣∣∣t=0

(f γ)(t).

(3) In spherical coordinates (r, θ, φ) in R3, ∂/∂r is the unit tangent vector tothe curve: θ = constant, φ = constant =⇒ ??? components of ∂/∂r in(x1, x2, x3).


Figure 2.5: Spherical coordinates

PROOF. Note that

φ1(x) = (r, θ, φ), φ2(x) = (x1, x2, x3), ai φ2(x) = xi

and φ2 φ−11 : (r, θ, φ) 7→ (x1, x2, x3) is given by

x1 = a1 φ2 φ−11 (r, θ, φ) = r sin θ cos φ,

x2 = a2 φ2 φ−11 (r, θ, φ) = r sin θ sin φ,

x3 = a3 φ2 φ−11 (r, θ, φ) = r cos θ.

Then(T(φ2 φ−1

1 )∂

∂r

)(h) =

∂

∂r(h φ2 φ−1

1 ) =∂h∂xi

∂(ai φ2 φ−11 )

∂r.

Hence∂

∂r∼= T(φ2 φ−1)

∂

∂r= sin θ cos φ

∂

∂x1 + sin θ sin φ∂

∂x2 + cos θ∂

∂x3 .

2.2.2. Fibre bundles. A bundle (E ,B, π,F ) consists of(i) topological spaces E and B,

(ii) continuous surjective mapping π : E → B such that ∀ x ∈ B the fiber atx, π−1(x) =: Ex, is homeomorphic to a topological space F .

We call B the base and F the typical fibre.


(1) A fibre bundle (E ,B, π,F ,G) is a bundle (E ,B, π,F ) together with atopological group G (called a a structure group) of homeomorphisms ofF onto itself, and an open covering (Ui)i∈I of B, such that

(1.1) locally the bundle is a trivial bundle:

π−1(Ui)φi∼=

//

π

Ui ×F

yysssssssssss

Ui

p //

π

(π(p),φ•i (p))

xxqqqqqqqqqq

π(p)

We call ((Ui,φi))i∈I a family of local trivializations of the bundle.For ∀ x ∈ Ui, φ•i,x := φ•i |Fx : Fx → F is homeomorphic.

(1.2) ∃ correlation of the trivial subbundles defined on the open covering(Ui)i∈I of B. ∀ x ∈ Ui ∩ Uj we have

φ•j,x φ•i,x ∈ G

(1.3) The induced mapping gij : Ui ∩ Uj → G given by

(2.2.16) gij(x) := φ•j,x φ•i,x, ∀ x ∈ Ui ∩ Uj

called the transition functions (see Figure 2.6), are continuous. Notethat

(2.2.17) gij(x)g jk(x) = gik(x), ∀ x ∈ Ui ∩ Uj ∩ Uk.

(2) A vector bundle is a fibre bundle (E ,B, π,F ,G) where F is a vectorspace and G is the linear group.

(3) A bundle morphism between two bundles (E1,B1, π1,F1) and (E2,B2, π2,F2)

is a pair of mappings (f, f) such that the diagram

E1f−−−−→ E2

π1

y yπ2

B1 −−−−→f

B2

is commutative, and f : E1,x → E2,f(x2)is linear for each x ∈ B1.

(4) A bundle category Bundle consists of bundles and bundle morphisms.(5) A fibre bundle (E ,B, π,F ,G) is said to be a Ck fibre bundle if E ,B,F are

Ck manifolds, π is a Ck mappings, G is a Lie group, and the covering ofB being the domains of an admissible atlas, the mappings gij are Ck.

(6) A chart (U ,φ) on E defines fibre coordinates on E if the mapping φ :U → Rn+p is a bundle morphism, with Rn+p having the natural bundlestructure Rn+p = Rn × Rp. Here (E ,B, π,F ,G) is a differential fibrebundle, dim E = n + p, dimB = n.

♣ Exercise: Show that Bundle is a category.

Tangent bundle. Let X be a differentiable manifold of dimension n. Define

(2.2.18) TX := (x, vx) : x ∈ X , vx ∈ TxX .


Figure 2.6: Transition functions

Then(TX ,X , π, Rn, GL(n, R))

is a fibre bundle.

(1) Fibre at x: TxX ; Typical fibre: Rn; Projection: π : TX → X , (x, vx) 7→ x.(2) Covering of X :

X =∪i∈IUi, where (Ui, ψi)i∈I is an atlas of X .

(3) Homomorphism φi:

π−1(Ui)φi //

π

Ui ×Rn

yysssssssssss

Ui

withφi := (π, ψ′i π2),

where π2(x, vx) = vx ∈ TxX and ψ′i(vx) = v is the representative of vxin the chart (Ui, ψi). The fibre coordinates on TX are given by

(ψi, 1) (π, ψ′i π2) : π−1(Ui) −→ Rn ×Rp = Rn+p,

p = (x, vx) 7−→ (x1, · · · , xn, v1, · · · , vn).(2.2.19)


(4) Structure group:

GL(n, R) = n× n real matrices A with det(A) = 0.

∀ x ∈ Ui ∩ Uj, we have

ψ′i,x : TxX → Rn, vx 7→ v and ψ′j,x : TxX → Rn, vx 7→ w

so that ψ′j,x ψ′−1i,x : Rn → Rn, v 7→ w.

(5) X is Ck =⇒ TX is Ck−1.

Frame bundle. Let X be a C∞-manifold.

(1) A frame ρx in TxX is a set of n linearly independent vectors (e1, · · · , en)which can be expressed as a linear combination of the elements of a par-ticular basis (e1 , · · · , en) of TxX :

ei = ajiej , (aj

i) ∈ GL(n, R).

Thenframes in TxX ←→ GL(n, R).

Let

(2.2.20) FX := (x, ρx) : x ∈ X and ρx is a frame at x.

Then(FX ,X , π, GL(n, R), GL(n, R))

is a fibre bundle.(2) A frame ρx can be thought of as a nonsingular linear mapping

ρx : Rn −→ TxX , (v1, · · · , vn) 7−→ vx := ∑1≤i≤n

viei

if ρx = (e1, · · · , en).

A fibre bundle (E ,B, π,F ,G) in which F and G are isomorphic and in whichG acts on F by left translation, is called a principal fibre bundle. For example,(FX ,X , π, GL(n, R), GL(n, R)) is a principal fibre bundle.

(1) Let (E ,B, π,F ,G) be a principal fibre bundle. Let (Ui)i∈I be the opencovering of X used to define the fibre bundle structure. Let p ∈ Ex andx ∈ Ui, define

(2.2.21) gi := φ•i,x(p)

where we identify F with G:

Exφ•i,x−−−−→∼=

F −−−−→∼= G

and ∀ g ∈ G,

(2.2.22) Rg p ≡(

Rg p)

i:= φ•−1

i,x (Rggi) = φ•−1i,x (gig), p ∈ π−1(Ui) =

∪x∈Ui

Ex.


Figure 2.8: (Rg p)i = (Rg p)j

Since Rg1 Rg2 p = Rg2g1 p, it follows that (Rg)g∈G is a group anti-isomorphicto G, acting on the right on π−1(Ui).

(Rg)g∈G ×π−1(Ui) −→ π−1(Ui),

(Rg, p ∈ Ex) 7−→ Rg p ∈ Ex

so (Rg)g∈G acts transitively on each fibre Ex, x ∈ π−1(Ui).(2) ∀ p ∈ π−1(Ui ∩ Uj) =⇒ ( see Figure 2.8)(

Rg p)

i=(

Rg p)

j.

PROOF. For p ∈ Ex with x ∈ Ui ∩ Uj =⇒

φ•i,x(p) = gi, φ•j,x(p) = gj, gi = φ•i,x φ•−1j,x (gj).

Then gi = gij(x)gj and(Rg p

)j= φ−1

j,x (gjg) = φ•−1i,x φi,x φ•−1

j,x (gjg)

= φ•−1i,x

(gij(x)gjg

)= φ•−1

i,x (gig) =(

Rg p)

i.

(3) Since the mapping Rg is independent of the choice of the open set Uicontaining π(p), it is well-defined over all of E , and we can write

(2.2.23) Rg(p) = φ•−1i,x Rg φ•i,x(p), x = π(p).


Hence we get the right action of G on E :

(2.2.24) E × G −→ E , (p, g) 7−→ pg := Rg(p).

Since p ∈ Ex =⇒ Rg(p) ∈ Ex, (2.2.24) is a fibre-preserving global right-action.

(4) Rg geometric meaning: connection. Rg : G → G ∼= F ∼= Ex =⇒TeRg : g = TeG → TpEx with π(p) = x.

(5) Examples:(5.1) In general, the left action of G on the principal fibre bundle (E ,X , π,F ,G)

does not define a fibre-preserving global action. Indeed, we, as thesame construction, obtain

gi = φ•i,x(p),(

Lg p)

i= φ•−1

i,x (Lggi) = φ•−1i,x (ggi).

Using φ•i,x φ•−1j,x (g) = gij(x)g yields gi = gij(x)gj and(

Lg p)

j= φ•−1

j,x (ggj) = φ•−1i,x φ•i,x φ•−1

j,x (ggj) = φ•−1i,x

(gij(x)ggj

)=(

Lg p)

i.

(5.2) G = GL(n, R) E = FX . Indeed, ∀ x ∈ Ui and ∀ p ∈ π−1(Ui) andlet φ•i,x(p) = gi = (a(i)

µλ) ∈ G ∼= F ∼= Ex. The action of G on the

typical fibre Ex is

Aµα a(j)

αλ = a(i)

µλ ⇐⇒ φ•i,x φ•−1

j,x = (Aµα)

and the right action of G on E is(Rg p

)i

= φ•−1i,x

((a(i)

µα Gα

λ

))= φ•−1

i,x

((Aµ

βa(j)βα Gα

λ

))= φ•−1

i,x φ•i,x φ•−1j,x

((a(j)

βα Gα

λ

))= φ•−1

j,x

((a(j)

βα Gα

λ

))with g = (Gα

β).

Given a fibre bundle (E ,B, π,F ,G), it it admits an equivalent structure de-fined with a subgroup G1 of G, then ∃ a family of local trivializations with transi-tion functions gij taking their value in G1. We say that G is reducible to G1.

(1) The structure group GL(n, R) of TRn is reducible to 1. In general, ∀fibre bundle with base Rn is reducible to a trivial bundle (since Rn is acontractible space).

(2) A principal fibre bundle (E ,X , π,F ,G) is said to be reducible to the prin-cipal fibre bundle (E1,X , π1,F1,G1) if

(2.1) G1 < G and E1 ⊂ E ,(2.2) the injection f : E1 → E is a bundle morphism commuting with the

action of G1:

G1 × E1Rg−−−−→ E1

1×fy yf

G1 × E −−−−→Rg

E


with

π(f(p)) = π1(p), ∀ p ∈ E1,

f(Rg(p)) = Rg(f(p)), ∀ p ∈ E1, ∀ g ∈ G1.

(3) FX is reducible to the bundle of orthogonal frames or Lorentz frames.

Theorem 2.2.2. A smooth principal fibre bundle (E ,X , π,F ,G) is reducible to(E1,X , π1,F1,G1) with G1 a Lie subgroup of the Lie group G ⇐⇒ ∃ family of localtrivializations whose transition functions take their value in G1.

2.2.3. Vector fields. Given a bundle (E ,B, π,F ).(1) A cross-section is a mapping f : B → E such that π f = 1B .

Theorem 2.2.3. A principal fibre bundle (E ,X , π,F ,G) is trivial⇐⇒ ∃ a continuouscross-section.

PROOF. (i) triviality =⇒ cross-section. Define

α : X −→ X × G, x 7−→ (x, f(x))

Then f is a continuous cross-section.(ii) cross-section =⇒ triviality. Let f : X → E be a continuous cross-section.

Given p ∈ E , ∃ x ∈ X such that p ∈ Ex, ∃! g0 ∈ G such that

p = Rg0(f(x)).

Define

α : E −→ X × G, p 7−→ (x, g0)

which preserves the group structure of the fibres:

α(Rg′ p) = α(

Rg0g′ f(x))= (x, g0g′) = (x, g0)g′ = Rg′α(p)

for all g′ ∈ G and p ∈ E . Note that α(f(x)) = (x, e), e ∈ G identity.

Let X be a Cr-manifold of dimension n, where r ∈ 1, 2, · · · ,+∞, ω.(1) A vector field v on X is a cross-section of (TX ,X , π, Rn).

(1.1) ∀ x ∈ X , the vector field v associates a tangent vector vx ∈ TxX .Thus

(2.2.25) v : X −→ TX , x 7−→ (x, vx) or vx.

(2) A Cr-vector field is a vector field v on X such that v : X → TX is Cr

(r ≤ k− 1).(2.1) A vector field v is Cr ⇐⇒ ∀ chart of an admissible atlas on X the n

functions vi are of class Cr.


(2.2) A vector field v can be defined as a derivation on Ck(X ):

(2.2.26) v : Ck(X ) −→ Ck−1(X ), f 7−→ v( f )

with vx( f ) = (v( f ))(x). Locally

(2.2.27) (v( f ))(x) = vx( f ) = vix

∂ f∂xi

∣∣∣∣x

=⇒ v f = vi ∂ f∂xi .

(3) Lie algebra X(X ) (where X is a smooth manifold of dimension n):

(2.2.28) X(X ) := C∞ vector fields on X.

Addition in X(X ):

(v + w)( f ) := v( f ) + w( f ), ∀ v, w ∈ X(X ), ∀ f ∈ C∞(X ).

Multiplication of v ∈ X(X ) by g ∈ C∞(X ):

C∞(X )×X(X ) −→ X(X ), (g, v) 7−→ gv, (gv)( f ) := g(v( f )).

Then X(X ) is a module on the ring C∞(X ). However, X(X ) is not closedunder multiplication defined by (vw)( f ) := v(w( f )):

(vw)( f g) = v(w( f g)) = v ( f (wg) + g(w f ))= f ((vw)g) + g ((vw) f ) + [(v f )(wg) + (vg)(w f )] .

Lie bracket on X(X ):

(2.2.29) [·, ·] : X(X ×X(X ) −→ X(X ), (v, w) 7−→ [v, w] := vw− wv =: Lvw.

Here Lvw is called the Lie derivative of w in the direction of v. Since

(vw) f = vi ∂

∂xi

(wj ∂ f

∂xj

)= viwj ∂2 f

∂xi∂xj + vi ∂wj

∂xi∂ f∂xj ,

we get

(2.2.30) [v, w] f =

(vi ∂wj

∂xi − wi ∂vj

∂xi

)∂ f∂xj ∈ X(X ).

Observe that: [·, ·] is distributive with respect to addition and anti-commutative,but is not associative. We have the Jacobi identity:

(2.2.31) [v1, [v2, v3]] + [v2, [v3, v1]] + [v3, [v1, v2]] = 0.

♣ Exercise: (i) Show that X(X ) is a module on the ring C∞(X ). (ii) Verify(2.2.31).

(4) A moving frame (if n = 4, vierbein or tetrad) is a set of n linearly in-dependent C∞ vector fields (ei)1≤i≤n which form a basis for the moduleX(U ), U ⊂ X .

(5) The image of a vector at x ∈ X under a differentiable map f : X → Y isgiven by

(f′vx)(g) := vx(g f), vx ∈ TxX , g ∈ C∞(x).

Then f′vx ∈ Tf(x)Y .


(5.1) When f is invertible, we get, for x = f−1(y) and v ∈ X(X ),[(f′v)(g)

](y) = [v(g f)](x) = [v(g f)](f−1(y))

so that(f′v)(g) = [v(g f)] f−1.

(5.2) v a Cr-vector field and f : X → Y a Cr+1-diffeomorphism =⇒ f′v isa Cr-vector field on Y .

(5.3) We say that a vector field v on X and a vector field w on Y are f-related if w = f′v.

Theorem 2.2.4. f : X → Y a C∞-diffeomorphism =⇒ f′ : X(X ) → X(Y) is anisomorphism of the Lie algebra. That is, f′[v, w] = [f′v, f′w].

PROOF. Compute(f′[v, w]

)(g) = ([v, w](g f)) f−1 = [v (w(g f))] f−1 − [w (v(g f))] f−1

and[f′v, f′w](g) = (f′v)

((f′w)g

)− (f′w)

((f′v)g

).

Because

(f′v)((f′w)g

)=

[v(((f′w)g) f

)] f−1

=[v((w(g f)) f−1 f

)] f−1 = [v (w(g f))] f−1.

Therefore (f′[v, w])(g) = [f′v, f′w](g).

(6) A vector field v on X is said to be invariant under the diffeomorphismf : X → X if

f′x(vx) = vf(x) ∀ x ∈ X ⇐⇒ f′v = v f.

Thenf∗((f′v)g

)= (f′v)g f = v(g f) = v(f∗g),

thusf∗ f′v = v f∗.

Hence

(2.2.32) v is f-invariant ⇐⇒ f∗ v f = v f∗.

2.2.4. Covariant vectors and cotangent bundles. Let X be an n-dimensionalsmooth manifold and x ∈ X .

(1) The dual T∗xX to TxX is the space of linear forms on TxX :

TxX = vx = tangent vectors/contravariant vectors,T∗xX := ωx = cotangent vectors/covectors/covariant vectors.∀ ωx ∈ T∗xX and vx ∈ TxX =⇒ ωx(vx) ∈ R.

(2) Given a base (ei)1≤i≤n in TxX , construct its dual (θi)1≤i≤n in T∗xX :

θi(vx) = θi(vjxej) = vj

xθi(ej) = vix, ⟨θi, ej⟩ = θi(ej) = δi

j.


(3) Let (dxi)1≤i≤n be the dual to the natural basis (∂/∂xi)1≤i≤n at x ∈ X :⟨dxi,

∂

∂xj

⟩= δi

j.

Hence for ωx = ωxidxi|x we get

⟨ωx, vx⟩ =⟨

ωxidxi|x, vjx

∂

∂xj

∣∣∣∣x

⟩= ωxiv

jxδi

j.

(4) Change of basis in T∗xX induced by a change of basis in TxX . If ei = aji ej,

then vx = vixei = vi

xaji ej = vj

x ej =⇒

θ j(vx) = θ j(vkxek) = vj

x = vixaj

i = ajiθ

i(vx) =⇒ θi = (a−1)ij θ

j.

In particular

(2.2.33) dxi =∂xi

∂xj

∣∣∣∣xdxj.

(5) A covariant vector at x ∈ X can also be defined as an equivalence classof triples (U ,φ, w), where (U ,φ) is a chart of X at x ∈ X and w ∈ Rn,the equivalence relation being defined by the transformation law givenabove.

d f |x(vx) = ⟨d f |x, vx⟩ = vx( f ) = vix

∂ f∂xi

∣∣∣∣x

so that

(2.2.34) d f (x) =∂ f∂xi dxi.

(6) ∀ x ∈ X and ∀ vx ∈ TxX , we have (ei → TxX and θi → T∗xX )

d f |x(vx) = ⟨d f |x, vixei⟩ = vi

x⟨d f |x, ei⟩ = vixei(g)

so that

(2.2.35) d f |x = ei( f )θi.

(7) Let

(2.2.36) T∗X := (x, ωx) : ωx ∈ T∗xXand

(T∗X ,X , π, Rn, GL(n, R))

the cotangent bundle.(8) A Cr covariant vector field or Cr one-form is a Cr cross-section of T∗X .

Let f : X → Y be a differentiable mapping.(1) The reciprocal image or pull-back of a covariant vector θy under a dif-

ferentiable mapping f is defined by

(2.2.37)((T∗xf)θy

)vx := θy ((Txf)vx) , y := f(x).

Then ⟨(T∗xf)θf(x), vx

⟩=⟨

θf(x), (Txf)vx

⟩.


Figure 2.9: Reciprocal image of a 1-form

(2) The reciprocal image of a 1-form θ under a differentiable mapping f is(see Figure 2.9)

(2.2.38) ((T∗f)θ) v = ⟨(T∗f)θ, v⟩ = ⟨θ, (Tf)v⟩ f = (θ((Tf)v)) f.

2.2.5. Tensors at a point. Let X be a Ck-manifold, dimX = n, and x ∈ X .The tensor product

Tp,qx X := ⊗pTxX

⊗⊗qT∗xX

of p tangent spaces at x and q cotangent spaces at x (the space of p-contravariantand q-covariant tensors) is the set of all multilinear forms on

T∗x × · · · × T∗xX︸︷︷︸p

×TxX × · · · × TxX︸︷︷︸q

.

A tensor in Tp,qx X is called a tensor of order type (p, q).

Theorem 2.2.5. Tp,qx X is a vector space of dimension np+q. In particular Tp,0

x X = TxXand T0,1

x X = T∗xX .

(1) (ei)1≤i≤n basis in TxX and (θi)1≤i≤n dual basis to (ei)1≤i≤n in T∗X =⇒ ∀v ∈ Tp,q

x X ,

v = vi1···ipj1···jq ei1 ⊗ · · · ⊗ eip ⊗ θ j1 ⊗ · · · ⊗ θ jq .


(2) Under a change of basis

ei = aji ej or ei = ai′

i ei′ ,

we haveθ j = aj

iθi or θi′ = ai′

i θi

so

(2.2.39) vi1···ipj1···jq = (a−1)i1

i′1· · · (a−1)

ipi′p

aj′1j1· · · aj′q

jq vi′1···i′pj′1···j′q

.

(3) Tensor algebra. ∀ u ∈ Tp,qx X and ∀ v ∈ Tr,s

x X , define the tensor productby

(2.2.40) (u⊗ v)i1···ip ···ip+rj1···jq ···jq+s

:= ui1···ipj1···jq v

ip+1···ip+rjq+1···jq+s

∈ Tp+r,q+sx X .

(4) Contraction. ∀ u ∈ Tp,qx X , ∀ 1 ≤ a ≤ p, ∀ 1 ≤ b ≤ q, define

(2.2.41)

Cab : Tp,q

x X −→ Tp−1,q−1x X , u =

(u

i1···ipj1···jq

)7−→ Ca

bu =

(∑

1≤k≤nu

i1···ia−1kia+1···ipj1···jb−1kjb+1···jq

).

(5) The symmetry properties of a tensor are the symmetry properties of thelinear form which defines it:

ω ∈ Tp,qx X ⇐⇒ ω : T∗xX × · · · × T∗xX︸︷︷︸

p

×TxX × · · · × TxX︸︷︷︸q

→ R.

Let Sq denote the set of all permutations of (1, · · · , q).(5.1) ∀ ω ∈ T0,q

x X define for π ∈ Sq,

(2.2.42) π(ω) ∈ T0,qx X ⇐⇒ (π(ω))(v1, · · · , vq) := ω(vπ(1), · · · , vπ(q)).

ω is symmetric (resp. anti-symmetric) defined by π if

π(ω) = ω (resp. π(ω) = (−1)|π|ω).

(5.2) ∀ ω ∈ Tp,0x X and ∀ π ∈ Sp, define

(2.2.43) (π(ω))(θ1, · · · , θp) := ω(θπ(1), · · · , θπ(p)).

(5.3) ∀ ω ∈ Tp,px X and ∀ π ∈ Sp, define

(2.2.44) (π(ω))(v1, · · · , vp, θ1, · · · , θp) := ω(vπ(1), · · · , vπ(p), θπ(1), · · · , θπ(p)).

(6) ∀ ω ∈ T0,qx X define

(6.1) The symmetrization operator of ω:

(2.2.45) S(ω) :=1q! ∑

π∈Sq

π(ω).

(6.2) The antisymmetrization operator of ω:

(2.2.46) A(ω) :=1q! ∑

π∈Sq

(−1)|π|π(ω).

Then(A(ω))j1···jq =

1q!

ϵi1···iqj1···jq ωi1···iq


where

(2.2.47) ϵi1···ipj1···jp

:=

0, (i1, · · · , iq) is not a permutation of (j1, · · · , jq),+1, (j1, · · · , jp) is an even permutation of (i1, · · · , ip),−1, (j1, · · · , jp) is an odd permutation of (i1, · · · , ip).

(7) q-forms at x := elements of T0,qx X .

2.2.6. Tensor bundles and tensor fields. LetX be a Ck-manifold of dimensionn.

(1) The tensor bundle of order (p, q) is

Tp,qX :=∪

x∈XTp,q

x X .

(2) A Cr-tensor field of order (p, q) is a Cr cross-section (r ≤ k− 1) of Tp,qX .(3) ∀ u, v ∈ Cr(X , Tp,qX ) and f ∈ Cr(X ), define

(u + v)i1···pj1···jq := u

i1···ipj1···jq + v

i1···ipj1···jq , ( f u)

i1···ipj1···jq (x) := f (x)u

i1···ipj1···jq (x).

(4) Let f : X → Y be Cr+1-differentiable, x ∈ X , y = f(x) ∈ Y =⇒

T0,qx f = ⊗qT∗xf : T0,q

y Y −→ T0,qx X , θy 7−→ (T0,q

x f)θy,

where

(2.2.48)((T0,q

x f)θy

)(v1, · · · , vq) := θy

((Txf)v1, · · · , (Txf)vq

).

(5) Let f : X → Y be a Cr+1-diffeomorphism, x ∈ X , y = f(x) ∈ Y =⇒

Tp,0x f = ⊗pTxf : Tp,0

x X −→ Tp,0y Y , vx 7−→ (Tp,0

x f)vx,

where

(2.2.49)((Tp,0

x f)vx

)(θ1, · · · , θp) := vx

((T∗xf)θ1, · · · , (T∗xf)θp

).

(6) From (4) and (5), we obtain, for a Cr+1-diffeomorphism f : X → Y ,

Tp,0f : Cr(X , Tp,0X ) −→ Cr(Y , Tp,0Y),

v 7−→((Tp,0f)v

)(y) =

(Tp,0

f−1(y)f)

vf−1(y),

T0,qf : Cr(Y , T0,qY) −→ Cr(X , T0,qX ),

θ 7−→((T0,qf)θ

)(x) :=

(T0,q

x f)

θf(x).

(7) Let f : X → Y be a Cr+1-diffeomorphism =⇒

Tp,0f−1 ⊗ T0,qf : Cr(Y , Tp,qY) −→ Cr(X , Tp,qX )

v⊗ θ 7−→(

Tp,0f(x)f

−1)

vf(x) ⊗(

T0,qx f)

θf(x).

Then[(Tp,0f−1 ⊗ T0,qf

)(v⊗ θ)(x)

](η1, · · · , ηp, w1, · · · , wq)

=[(

Tp,0f(x)f

−1)

vf(x) ⊗(

T0,qx f)

θf(x)

](η1, · · · , ηp, w1, · · · , wq)

= vf(x)

((T∗f(x)f

−1)η1, · · · , (T∗f(x)f−1)ηp

)· θf(x)

((Txf)w1, · · · , (Txf)wq

).

2.3. LIE GROUPS 87

2.3. Lie groups

2.3.1. Groups of transformations. Let X be a smooth manifold of dimensionn.

(1) v vector field =⇒ σ : R ⊃ I → X is an integral curve of v if σ is a curvesuch that

(2.3.1) σ(t) :=dσ(t)

dt= v(σ(t)) ∈ Tσ(t)X , ∀ t ∈ I.

Theorem 2.3.1. v is a Cr vector field =⇒ ∀ x ∈ X ∃ integral curve of v, t 7→ σ(t, x),such that

(i) σ(t, x) is defined for t ∈ Ix ⊂ R and is of class Cr+1, where Ix is an integralcontaining 0,

(ii) σ(0, x) = x, ∀ x ∈ X ,(iii) this curve is unique: Given x ∈ X ∃/ C1 integral curve of v defined on an

interval strictly greater than Ix, and passing through x (i.e., such that σ(0, x) =x).

PROOF. Use Theorem 1.9.1 and Theorem 1.9.2.

Theorem 2.3.2. In Theorem 2.3.1, if t, s, t + s ∈ Ix, then σ(t, σ(s, x)) = σ(t + s, x).

PROOF. As an exercise.

♣ Exercise: Complete the proof of Theorem 2.3.2.

(2) Let

(2.3.2) Σv := (x, t) ∈ X ×R : x ∈ X and t ∈ Ixwhich is open in X ×R, with dimension n + 1.

(3) The flow of the C1 vector field v is the mapping

(2.3.3) σ : Σv −→ X , (x, t) 7−→ σ(t, x).

(3.1) X and v are smooth =⇒ σ is C∞.(3.2) X is not compact =⇒ Σv is not a product X × I.(3.3) ∀ x0 ∈ X ∃ neighborhood Nx0 ⊂ X and then interval Ix0 ⊂ R such

that σ is defined on Nx0 × Ix0 , and smooth if X and v are smooth.(4) The mapping, ∀ t ∈ Ix0 ,

(2.3.4) σt : Nx0 −→ X , x 7−→ σt(x) := σ(t, x).

Let

I := the intersection of all the intervals Ix0 correspondingto a set of neighborhoods (Nx0)x0 that covers X

Then I = ∅ or I = ∅.(4.1) X compact =⇒ I = ∅.


Figure 2.9: Flows

(4.2) I = ∅ =⇒ (σt)t∈I defines a global transformation of X .(4.3) σt+s = σt σs and σ−1

t = σ−t, ∀ t ∈ R.

Theorem 2.3.3. A C∞ vector field on a compact manifold X generates a one parametergroup of diffeomorphisms of X .

Corollary 2.3.4. A C∞ vector field on a manifold X , which vanishes outside a compactset K ⊂ X , generates a one parameter group of diffeomorphisms of X .

PROOF. ∀ x ∈ X \ K, σt is the identity mapping on Nx, ∀ t ∈ R. By coveringK with a finite number of open neighborhoods Nx, ∃ ∅ = I ⊂ R such that σt isdefined on X for all t ∈ I, thus for all t ∈ R.

(5) The Set of mappings (σt)t∈I onX is called a one parameter local pseudo-group if

σt σs = σt+s, ∀ t, s, t + s ∈ I.

(5.1) It is a one parameter local group if (σt)t∈I defines a global transfor-mation of X .

(5.2) It is a one parameter group if it is a one parameter local group withI = R.

2.3. LIE GROUPS 89

Figure 2.10: Theorem 2.3.5

(5.3) ∀ one parameter local pseudo-group (σt)t∈I of transformations σt :x 7→ σt(x) = σ(t, x) can be generated by a vector field v defineduniquely by the equation v(x) := (d/dt|t=0)σ(t, x):

v(x) =ddt

∣∣∣∣t=0

σ(t, x) =⇒ v(σ(t, x)) =ddt

σ(t, x).

PROOF. For t, s, t + s ∈ I, we have

ddt

σ(t + s, x) =dds

σ(t + s, x) =dds

σ(s, σ(t, x))

so that

ddt

σ(t, x) =dds

∣∣∣∣s=0

σ(s, σ(t, x)) = v(σ(t, x)).

Such a v is called the infinitesimal generator.

Theorem 2.3.5. Given a vector field v generating the local pseudo-group (σt)t∈I , theimage f′v of v under the diffeomorphism f of X onto X generates the one parameter localpseudo-group (f σt f−1)t∈I .

PROOF. Set

w(y) :=ddt

∣∣∣∣t=0

(f σt f−1)(y).

Then, for x = f−1(y),

w(y) =ddt

∣∣∣∣t=0

(f σt)(x) = f′x

(ddt

∣∣∣∣t=0

σt(x))= f′f−1(y)(vf−1(y)) = (f′v)(y).

Hence w = f′v.


Figure 2.11: Lvw.

Corollary 2.3.6. The vector field v is invariant under f (a diffeomorphism on X )⇐⇒ fcommutes with σt.

PROOF. v is invariant under f⇐⇒ f′v = v. But

(f′v)y =ddt

∣∣∣∣t=0

(f σt f−1)(y), v(y) =ddt

∣∣∣∣t=0

σt(y).

We see that v is invariant under f if and only if f σt f−1 = σt.

2.3.2. Lie derivatives. The flow of a vector field v determines local transfor-mations σt of a smooth manifold X , mappings of neighborhoods Nx ⊂ X intoX .

(1) The Lie derivative Lvw of the contravariant vector field w is the con-travariant vector field defined by (see Figure 2.11)

(2.3.5) Lvw∣∣∣∣x

:= limt→0

(Tσt(x)σ−1t )wσt(x) − wx

t.

(2) The Lie derivative Lvω of the covariant vector field ω is the covariantvector field defined by (see Figure 2.12)

(2.3.6) Lvω

∣∣∣∣x

:= limt→0

(T∗xσt)ωσt(x) −ωx

t.

(3) ∀ function f =⇒

(2.3.7) Lv f∣∣∣∣x= lim

t→0

f (σt(x))− f (x)t

= vx f = (v f )x.

Proposition 2.3.7. (i) The Lie derivative is a local operator:(a) u1 = u2 on Nx ⊂ X =⇒ Lvu1 = Lvu2 on Nx.(b) v1 = v2 on Nx ⊂ X =⇒Lv1 u = Lv2 u on Nx.

2.3. LIE GROUPS 91

Figure 2.12: Lvω.

(ii) The Lie derivative is a derivation on the algebra of (germs of) differentiable tensorfields, i.e.,

(c) Lv is an additive operator,

Lv(u1 + u2) = Lvu1 +Lvu2.

(d) Lv satisfies Leibniz’s rule

Lv(u1 ⊗ u2) = Lvu1 ⊗ u2 + u1 ⊗Lvu2.

PROOF. We only prove (d) in the case that u1, u2 are both vector fields. Forany ω1, ω2 ∈ T∗xX ,

Lv(u1⊗ u2)

∣∣∣∣x(ω1, ω2) =

(limt→0

(Tσt(x)σ−1t )(u1 ⊗ u2|σt(x))− u1 ⊗ u2|x

t

)(ω1, ω2)

= limt→0

(Tσt(x)σ−1t )(u1 ⊗ u2|σt(x))(ω1, ω2)− u1|x(ω1)u2|x(ω2)

t

= limt→0

u1|σt(x)((T∗σt(x)σ−1t )(ω1))u2|σt(x)((T∗σt(x)σ

−1t )(ω2))− u1|x(ω1)u2|x(ω2)

t

= limt→0

u1|σt(x)((T∗σt(x)σ−1t )(ω1))− u1|x(ω1)

t

u2|x(ω2)

+ limt→0

u1|σt(x)

((T∗σt(x)σ

−1t

)(ω1)

)u2|σt(x)((T∗σt(x)σ−1t )(ω2))− u2|x(ω2)

t

= Lvu1

∣∣∣∣x(ω1) · u2|x(ω2) + u1|x(ω1) ·Lvu2

∣∣∣∣x(ω2)

= (Lvu1 ⊗ u2 + u1 ⊗Lvu2)x (ω1, ω2).

Thus Lv(u1 ⊗ u2) = Lvu1 ⊗ u2 + u1 ⊗Lvu2.

♣ Exercise : Complete the proof of Proposition 2.3.7.


(4) Local coordinate expressions of Lie derivatives. For example

T = Tijk(x)

∂

∂xi ⊗ dxj ⊗ dxk,

then

LvT∣∣∣∣x

=(LvTi

jk

)x

∂

∂xi ⊗ dxj ⊗ dxk + Tijk(x)Lv

(∂

∂xi

)⊗ dxj ⊗ dxk

+ Tijk(x)

∂

∂xi ⊗Lv(dxj)⊗ dxk + Tijk(x)

∂

∂xi ⊗ dxj ⊗Lv(dxk).

Compute, using δji = ∂pj(0, y)/∂yi and ∂/∂xi|y = ∂/∂yi|y,(

Lv∂

∂xi

)x

= limt→0

1t

[(Tyσ−1

t

)( ∂

∂xi

∣∣∣∣y

)− ∂

∂xi

∣∣∣∣x

]

= limt→0

1t

[∂pj(t, y)

∂yi − δji

] (∂

∂xj

∣∣∣∣x

)=

(ddt

∣∣∣∣t=0

∂pj(t, y)∂yi

)∂

∂xj

∣∣∣∣x

= − ddt

∣∣∣∣t=0

∂σj(t, x)∂xi · ∂

∂xj

∣∣∣∣x

(ddt

(∂pj

∂yi∂σk

∂xj

)=

ddt

δki = 0

)

= − ∂

∂xi vj(x) · ∂

∂xj

∣∣∣∣x.

Hence

(2.3.8) Lv∂

∂xi = −∂vj

∂xi∂

∂xj .

Similarly

(2.3.9) Lvdxi =∂vi

∂xj dxj.

In summary,

(LvT)ijk = vℓ∂ℓTi

jk − Tℓjk∂ℓvi + Ti

ℓk∂jvℓ + Tijℓ∂kvℓ.

In particular, ∀ vector field w,

Lvw = Lv

(wi ∂

∂xi

)=

(Lvwi

) ∂

∂xi + wiLv∂

∂xi = (vwi)∂

∂xi − wi ∂vj

∂xi∂

∂xj

=

[vj ∂wi

∂xj − wj ∂vi

∂xj

]∂

∂xi = [v, w]i∂

∂xi = [v, w].

Theorem 2.3.8. A tensor field ω is invariant under a (local, pseudo) group (σt)t∈I oftransformations generated by v⇐⇒Lvω = 0.

(6) A metric g is a 2-covariant nondegenerate symmetric tensor field on X .(6.1) An isometry is a diffeomorphism of X which leaves g invariant.(6.2) Let

(2.3.10) Iso(X , g) := isometries of (X , g) .

Observe that Iso(X , g) is a group.

2.3. LIE GROUPS 93

Figure 2.13: Local coordinate expressions of Lie derivatives.

(6.3) A one parameter group of transformations generated by a vectorfield v is a group of isometries⇐⇒ 0 = Lvg. Because

(Lvg)ij = vk(

∂gij

∂xk

)+ gik

(∂vk

∂xj

)+ gkj

(∂vk

∂xi

)

=∂vi

∂xj +∂vj

∂xi + vk(

∂gij

∂xk −∂gik

∂xj −∂gkj

∂xi

)with vi := gijvj. We will prove that

0 = Lvg ⇐⇒ 0 = ∇ivj +∇jvi

where ∇ is the Levi-Civita connection of g.We call Lvg the stain tensor field generated by v.

2.3.3. Lie group of transformations. Let X be a smooth manifold of dimen-sion n and G a Lie group of dimension p.

(1) The set (σg)g∈G is a Lie group of transformations if the mapping

(2.3.11) σ : G × X −→ X , (g, x) 7−→ σ(g, x) = σg(x)

is smooth and if the set of transformations (σg)g∈G on X satisfies(1.1) σg σh = σgh (left action of G on X ) or σg σh = σhg (right action

of G on X ),(1.2) σe = 1X (hence σ−1g = σg−1 ).

(2) G operates(2.1) effectively on X if (σg(x) = x for all x ∈ X =⇒ g = e);(2.2) freely on X if (∀ g = e, ∀ x ∈ X =⇒ σg(x) = x);


Figure 2.14: Lie group of transformations.

(2.3) transitively on X if (∀ x, y ∈ X ∃ g ∈ G such that σg(x) = y).(3) A one parameter subgroup of a Lie group G is a smooth curve

R −→ G, t 7−→ g(t)

such that

g(0) = e and g(t)g(s) = g(t + s).

(4) The concepts developed for (σt)t∈I apply to (σg(t))t∈R:

(5) The vector field which generates the group of transformations (σg(t))t iscalled a Killing vector field on X relative to the action of G.

(5.1) The integral curve through x of the Killing vector field v satisfies

(2.3.12)dσx(g(t))

dt= v(σx(g(t))), σx(e) = x.

(5.2) We shall see that a one-parameter subgroup is defined by

(2.3.13) γ :=ddt

∣∣∣∣t=0

g(t) ∈ T)eG ∼= g.

Hence the Killing vector field generating (σg(t))t∈R by γ can be rela-beled by

(2.3.14) vγ(x) :=ddt

∣∣∣∣t=0

σg(t)(x) =ddt

∣∣∣∣t=0

σx(g(t)) = (Teσx)(γ) ∈ TxX .

(5.3) G acts effectively on X =⇒ Killing vector fields ∼= g.

2.3. LIE GROUPS 95

Figure 2.15: Group transformations.

(5.4) Group transformations:

GLg−−−−→ G

Rg

yG

Here Lg(h) := gh (left transformation) and Rg(h) = hg (right trans-formation).

(6) Notation: x ∈ X with coordinates (xi)1≤i≤n in Rn and g ∈ G with coor-dinates (gα)1≤α≤p in Rp. 1 ≤ i, j, k, ℓ, · · · ≤ n and 1 ≤ α, β, γ, δ, · · · ≤ p.

(6.1) σ : G ×X → X , (g, x) 7→ σ(g, x) with coordinates (σi(gα, xj)). Then

σg : X −→ X , x 7−→ σ(g, x),σx : G −→ X , g 7−→ σ(g, x),

and

Txσg : TxX −→ Tσg(x)X , (Txσg)ij :=

∂σi(gα, xk)

∂xj ,

Tgσx : TgG −→ Tσx(g)X , (Tgσx)iα :=

∂σi(gβ, xj)

∂gα.

(6.2) If the group of transformations operates on the group itself, then

G × G L−−−−→R

G,


and

L(g, h) = gh = Lg(h) (gh)α = Lα(gβ, hγ),

R(g, h) = hg = Rg(h) (hg)α = Rα(gβ, hγ),

and

ThLg : ThG −→ TghG, (ThLg)αδ =

∂Lα(gβ, hγ)

∂hδ,

ThRg : ThG −→ ThgG, (ThRg)αδ =

∂Rα(gβ, hγ)

∂hδ.

Let G be a Lie group of dimension p.(1) Lg : g ∈ G and Rg : g ∈ G act effectively, transitively, freely on G.(2) A vector field v on G is left invariant if

(2.3.15) (ThLg)(vh) = vLg(h) = vgh, ∀ g, h ∈ G.

A vector field v on G is right invariant if

(2.3.16) (ThRg)(vh) = vRg(h) = vhg, ∀ g, h ∈ G.

♣ Exercise: Show that Lg : g ∈ G and Rg : g ∈ G act effectively,transitively, freely on G.

Theorem 2.3.9. ∃ bijective correspondence between left invariant vector fields andvector tangent to G at e = TeG.

PROOF. Define

α : left invariant vector fields −→ TeG, v 7−→ ve.

Conversely, ∀ γ ∈ TeG, define v ≡ β(γ) ∈ X(G) by

vg := (TeLg)(γ), ∀ g ∈ G.

Then

vLg(h) = vgh = (TeLgh)(γ) =(Te(Lg Lh)

)(γ)

=((ThLg) (TeLh)

)(γ) = (ThLg)(vh).

Moreover, ∀ γ ∈ TeG,

(α β)(γ) = α(v) = ve = (TeLe)(γ) = γ

and ∀ left invariant vector field v,

((β α)(v))g = (β(ve))g = (TeLg)(ve) = vg, ∀ g ∈ G.

Thus (β α)(v) = v.

2.3. LIE GROUPS 97

Theorem 2.3.10. The set of left (right) invariant vector fields is closed under the Liebracket operation.

PROOF. If v and w are left invariant, then

(TLg) [v, w] = [(TLg)v, (TLg)w] = [v Lg, w Lg] = [v, w] Lg.

Similarly, the same argument holds for right invariant vector fields. (3) Lie algebra g = Lie(G) of G:

g := left invariant vector fields on G.(4) Let (vα)1≤α≤p be a basis of g =⇒ ∃ constants cγ

αβ satisfy

(2.3.17) [vα, vβ] := cγαβvγ.

We call cγαβ the structure constants of the Lie group G.

(4.1) Anti-symmetry: cγαβ = −cγ

βα.(4.2) Bilinearity:(4.3) Jacobi identity: cγ

αβcκγσ + cγ

βσcκγα + cγ

σαcκγβ = 0.

♣ Exercise: Verify the Jacobi identity.

(5) If vα(e) = ∂/∂gα|g=e is a basis of TeG ∼= g, then

vα(g) := (TeLg)(vα(e)), vκα(g) =

∂Lκ(g, h)∂hµ

∣∣∣∣h=e

γµα with γ

µα := vµ

α(e),

is a basis of g =⇒[vα, vβ](g) = cγ

αβvγ(g).

Therefore

cγαβvλ

γ(g) = vκα(g)

∂vλβ(g)

∂gκ− vκ

β(g)∂vλ

α (g)∂gκ

= γκαγ

µβ

∂2Lλ(g, h)∂gκ∂hµ

∣∣∣∣g=h=e

− γµα γκ

β

∂2Lλ(g, h)∂gκ∂hµ

∣∣∣∣g=h=e

(2.3.18)

=(

γµβγκ

α − γµα γκ

β

) ∂2

∂gκ∂hµ

∣∣∣∣g=h=e

Lλ(g, h).

(6) Structure constants of GL(n, R). g = (gab)1≤a,b≤n ∈ GL(n, R) =⇒

L(g, h) = (gh)ab = ga

c hcb.

If vab(e) := ∂/∂ga

b|g=e, then

(vab(g))i

j =∂(gh)i

j

∂hcd

∣∣∣∣h=e

(vab(e))

cd = gi

aδjb.

Therefore

(2.3.19) [vba(g), vd

c (g)] = δda vb

c(g)− δbc δd

a (g).


♣ Exercise: Verify (2.3.19).

Theorem 2.3.11. A Lie group G had vanishing structure constants ⇐⇒ it is locallyisomorphic to Rn (i.e., if and only if it is Abelian).

PROOF. ⇐=: On Rn,

(Lgh)α = (gh)α = gα + hα, (TeLg)αβ =

∂(Lgh)α

∂hβ

∣∣∣∣h=e

= δαβ

that is independent of g. Hence cγαβ = 0.

=⇒: See later. 2.3.4. One parameter subgroups. Let G be a Lie group of dimension p.

Theorem 2.3.12. The one parameter subgroups of G are the integral curves, through e, ofthe left (or right) invariant vector fields.

PROOF. Let g : R → G be a one parameter subgroup of G. From Lg(t)g(s) =g(t + s), we obtain(

Tg(s)Lg(t)

) dg(s)ds

=dg(t + s)

ds=

dg(t + s)dt

.

Setting s = 0 yields (TeLg(t)

)γ =

dg(t)dt

where γ is the tangent vector at g(0) = e to the curve t 7→ g(t). Moreover g(t) isan integral curve of the left invariant vector field equal to γ at e.

Conversely, given a left invariant vector field v, determined by γ = ve ∈ TeG,by Theorem 2.3.1, ∃! integral curve t 7→ σ(t, x) = σt(x) such that

dσ(t, x)dt

= v(σ(t, x)), σ(0, x) = x

for any given x ∈ G. Defineg(t) := σ(t, e).

The invariance implies that g(t) is defined for all t ∈ R, and then g(t)g(s) =g(t + s) for all t, s ∈ R by the uniqueness.

Let G be a Lie group and γ ∈ TeG.(1) Define v ∈ g, a left invariant vector field determined by γ = v(e),

(2.3.20) vg := (TeLg)(γ).

by Theorem 2.3.1, ∃! (g(t))t∈R such that

dg(t)dt

= v(g(t)) with g(0) = e =⇒ γ =ddt

∣∣∣∣t=0

g(t).

2.3. LIE GROUPS 99

Figure 2.16: vL,γ and vR,γ

(3) Let

(2.3.21) vL,γ(h) :=ddt

∣∣∣∣t=0

(Lg(t)h) =⇒ vL,γ(h) = (TeRh)(γ).

Similarly, define

(2.3.22) vR,γ(h) :=ddt

∣∣∣∣t=0

(Rg(t)h) =⇒ vR,γ(h) = (TeLh)(γ) = vh.

Consequently

left invariant vector fields ←→ TeG ∼= g←→ right invariant vector fieldswith

vL,γ ←→ γ←→ vR,γ.

Theorem 2.3.13. Let vL be the generator of the one parameter group of transformations(Lg(t))t∈R of G onto G =⇒ vL is a right invariant vector field. Similarly, the generatorvR of (Rg(t))t∈R is a left invariant vector field.

PROOF. vL(h) = ddt |t=0Rh(g(t)) = (TeRh)(

ddt |t=0g(t)).

Remark 2.3.14. vL (resp. vR) is a Killing vector field on G relative to the action L(resp. R) of the group.


2.3.5. Exponential mappings. Let G be a Lie group of dimension p and γ ∈TeG.

(1) ∃! integral curve t 7→ σ(t, e) such that

dσ(t, e)dt

= v(σ(t, e)) with σ(0, e) = e, vg := (TeLg)(γ) ∈ TgG.

Let

(2.3.23) gγ(t) := σ(t, e) ∈ G, t ∈ R.

(2) Define the exponential mapping by

(2.3.24) exp : TeG −→ G, γ 7−→ exp(γ) := gγ(1).

Then

exp(tγ) = gtγ(1) = gγ(t),exp(tγ) exp(sγ) = exp((t + s)γ) = gγ(t)gγ(s) = gγ(t + s).

(3) TeG ∼= g =⇒

gexp //

AAA

AAAA

A G

TeG

OO vexp //

%%KKKKKKKKKKK exp(γ)

γ := v(e) = ve

OO

(4) More generally, the solution of the differential equation

dσ(t, x)dt

= v(σ(t, x)), σ(0, x) = x

is written as

(2.3.25) σ(t, x) = expx(tv) = (exp(tv))x , ∀ v ∈ TxX .

The exponential mapping expx : TxX → X , where X is a Riemannianmanifold, maps line tw, |t| < ϵ, into the geodesic curve through x withtangent vector w at x.

Proposition 2.3.15. f ∈ Cω(G) =⇒ f admits the following expansion

(2.3.26) f (h exp γ) = ∑n≥0

1n!(vn f )(h), ∀ h ∈ G, ∀ γ ∈ TeG,

where v is the left invariant vector field generated by γ.

PROOF. Compute

(v f )(h) = vh f = ((TeLh) f ) ( f ) = γ( f Lh)

=ddt

∣∣∣∣t=0

( f Lh gγ(t)) =ddt

∣∣∣∣t=0

f (h exp(tγ)) .

2.3. LIE GROUPS 101

Figure 2.17: Proposition 2.3.15

Here gγ(t) was defined in (2.3.23). Hence

(v f ) (h exp(sγ)) =ddt

∣∣∣∣t=0

f (h exp(sγ) exp(tγ))

=ddt

∣∣∣∣t=0

f(

h exp(sγ)ddt

∣∣∣∣t=0

exp(tγ))

=dds

f (h exp(sγ))

and by induction

(vn f ) (h exp(sγ)) =dn

dsn [ f (h exp(sγ))] .

f is analytic =⇒

f (h exp(sγ)) = ∑n≥0

sn

n!dn

dsn

∣∣∣∣s=0

f (h exp(sγ)) = ∑n≥0

sn

n!(vn f )(h).

Thus we get (2.3.26).

(5) It can be shown that

exp(tα) exp(tβ) = exp

t(α + β) +12

t2[α, β]

+112

t3 ([α, [α, β]] + [[α, β], β]) + O(t4)

.(2.3.27)

(6) Let gγ(t) ≡ g(t, γ) be the one parameter subgroup generated by γ ∈ TeG.Let (eα)1≤α≤p be a basis in TeG =⇒ γ = γαeα. The canonical (normal)coordinates of g(t, γ) = exp(tγαeα) are

(2.3.28) gα(t, γ) = tγα, 1 ≤ α ≤ p.

(7) Given an arbitrary system of coordinates in G, ∃ a canonical system ofcoordinates, which is unique modulo a change of basis in TeG.


PROOF. Consider

g(t, aγ) = exp(taγ) = exp(atγ) = g(at, γ).

Let

i, j, k, ℓ, · · · coordinates in an arbitrary system,α, β, γ, δ, · · · coordinates in the canonical system.

Then

(exp(tγ))i = gi(t, γα) = gi(1, tγα) = gi(1, gα)

and

vi(exp(tγ)) = (vxi)(exp(tγ)) =ddt

xi (exp(tγ)) =ddt

gi(t, γα) =∂gi(1, gα)

∂gβγβ.

Setting t = 0 yields

γi =∂gi

∂gβ(1, 0)γβ;

thus ∂gi/∂gα(1, 0) = δiα.

The domain of validity of the transformations gi(1, gα) determinedby the range I × N0 ⊂ R × TeG on which the function g(t, γ) is de-fined and differentiable, can be characterized by N0 without referringto I, since g(1, γ) = g(a, a−1γ) and with a proper choice ofN0, where a issmall enough to be in I.

2.3.6. Lie groups of transformations. Let (σg)g∈G be a group of transforma-tions (diffeomorphisms) of a smooth manifold X (dimX = n, dimG = p, G a Liegroups).

(1) The mappingG −→ (σg)g∈G , g 7−→ σg

is called a realization of G. A realization defines an action of G on X :

G × X −→ X , (g, x) 7−→ σg(x) =: g · x (σg σh = σgh),(2.3.29)

X × G −→ X , (x, g) 7−→ σg(x) =: x · g (σg σh = σhg).(2.3.30)

We call them left action and right action.(2) Examples of left actions:

(2.1) The left translations, Lg : G → G, h 7→ Lg(h) = gh,(2.2) The inverse right translations: R−1

g (h) = Rg−1(h) = hg−1. Indeed,define

σ : G × G −→ G,(g, h) 7−→ σg(h) = σ(g, h) := hg−1.

Then

σg σh(k) = σg(kh−1) = kh−1g−1 = k(gh)−1 = σgh(k).

(2.3) The inner automorphisms,

(2.3.31) adg(h) := Lg R−1g (h) = ghg−1.

(3) If the mapping g 7→ σg is injective, the realization is faithful.

2.3. LIE GROUPS 103

(4) When the transformations σg : X → X are linear transformations of avector space X , the homomorphism: G → (σg)g∈G is called a represen-tation of G.

(5) ∀ γ ∈ TeG defines the generator vL,γ (resp. vR,γ) of a one parameter sub-group of transformations (Lgγ(t))t∈R (resp. transformations (Rgγ(t))t∈R)of G,

(2.3.32) vL,γ(h) = (TeRh)γ(

resp. vR,γ(h) = (TeLh)γ)

.

(6) ∀ γ ∈ TeG defines the Killing vector field vK,γ on X which generates thegroup of transformations (σgγ(t))t∈R of X :

(2.3.33) vK,γ(x) := (Teσx)γ, σx(gγ(t)) := σgγ(t)(x).

(7) Then

(2.3.34) dim(

vK,γ|γ ∈ TeG)

= rank(Teσx) =: r ≤ p = dimG.

Theorem 2.3.16. TFAE:(i) r = p.

(ii) vK,γ = 0⇐⇒ γ = 0.(iii) G acts effectively on X .(iv) vK,γ|γ ∈ TeG ∼= TeG.

PROOF. We only prove that (iii) =⇒ (i). Otherwise r < p. ∃ Killing vector fieldvK,γ on X , 0 = γ ∈ TeG, which is identically zero, say the Killing vector field isgenerated by (σgγ(t))t∈R. Let x(t) be the curve on X generated by (σgγ(t)(x0))t∈R:

vK,γ(x(t)) =dσgγ(t)(x0)

dt.

Then vK,γ ≡ 0⇐⇒ σgγ(t)(x0) = σgγ(0)(x0) = σr(x0) = x0 for all x0 ∈ X . HenceG does not act effectively on X .

(8) Determine the structure constants of G from the Lie bracket in the spaceof Killing vector fields vK,γ|γ ∈ TeG of a faithful realization of G on X .

PROOF. Faithful realization =⇒ G acts effectively on X . Theorem2.3.16 =⇒ vK,γ|γ ∈ TeG ∼= TeG ∼= g and vK,γ = 0 if and only if γ = 0.Let (v(α)(e))1≤α≤p be a basis in TeG. Let

v(α)(e) :=ddt

∣∣∣∣t=0

g(α)(t)

define vK(α) as well as the generators vL

(α) and vR(α) of the one parameter

subgroups (Lg(t))t∈R and (Rg(t))t∈R of transformations of G:

vK(α) := vK,v(α)(e), vL

(α) := vL,v(α)(e), vR(α) := vR,v(α)(e), g(α)(t) := gv(α)(e)(t).


We claim[vR(α), vR

(β)

]= cγ

αβvR(γ),

[vL(α), vL

(β)

]= −cγ

αβvL(γ),[

vK(α), vK

(β)

]= −cγ

αβvK(γ),

[vR−1

(α) , vR−1

(β)

]= −cγ

αβvR−1

(γ) ,

where vR−1

(α) is the generator of (R−1g(α)(t)

)t∈R. Using

σ(g, σ(h, x)) = σ(gh, x)

yields∂σi(g, σ(h, x))

∂gα=

∂σi(gh, x)∂(gh)β

∂(gh)β

∂gα.

Setting g = e we obtain

∂

∂gα

∣∣∣∣g=e

σi(g, σ(h, x)) =∂σi(h, x)

∂hβ

∂

∂gα

∣∣∣∣g=e

(gh)β.

The components of vL(α) are

vLβ

(α)(h) =

((TeRh) (v(α)(e))

)β=

∂

∂gα

∣∣∣∣g=e

(g(α)h

)β, g(α) := g(α)(e)

so that

vKi(α)(σ(h, x)) =

∂σi(h, x)∂hβ

vLβ

(α)(h).

Differentiating it with respect to hγ and setting h = e yields[vK(γ), vK

(α)

]i(x) = vKi

(β)(x)[vL(γ), vL

(α)

]β(e).

Let

σ : G × G −→ G, (g, h) 7−→ σ(g, h) := Rg−1(h) = hg−1 = R−1g (h).

Then σ(g, σ(h, k)) = σ(gh, k) and[vR−1

(γ) , vR−1

(α)

](k) = vR−1δ

(β) (k)[vL(γ), vL

(α)

]β(e).

On the other hand,

vR−1

(β) (k) =ddt

∣∣∣∣t=0

R−1g(β)(t)

(k) =ddt

∣∣∣∣t=0

Lk

(g−1(β)

(t))= −vR

(β)(k)

which implies[vR−1

(γ) , vR−1

(α)

]δ(k) = −vRδ

(β)(k)[vL(γ), vL

(α)

]β(e) = cβ

γαvRδ(β)(k) = −cβ

γαvR−1δ(β) (k).

Thus [vR−1

(γ) , vR−1

(α) ] = −cβγαvR−1

(β) .

Let G act effectively on X .

(1) vK,γ|γ ∈ TeG ∼= vL,γ|γ ∈ TeG ∼= vR,γ|γ ∈ TeG ∼= g.

2.3. LIE GROUPS 105

Theorem 2.3.17. The Killing vector fields on X defined by a left (resp. right) effectiveaction of G on X is isometric to the Lie algebra of the generators of left (resp. right) actionsof G on G.

PROOF. Since[vK(γ), vK

(α)

]i(x) =

∂σi(h, x)∂hβ

∣∣∣∣h=e

[vL(γ), vL

(α)

]β(e),

it follows that it is isometric.

(2) Given a matrix representation of G on Rn:

σg = [Dij(g)] : Rn −→ Rn, x 7−→ y = D(g)x.

Then[vK(α), vK

(β)

]j(x) =

(Dj

i,β(e)Dik,α(e)− Dj

i,αDik,β(e)

)xk = −

[D,α, D,β

]ik (e)xk,

where D,α(e) := ∂D(g)/∂gα|g=e ∈ TeG.

Invariance under a group of transformations. A tensor field on X is invariantunder a connected group G of transformations (σg)g∈G ⇐⇒ it is invariant underits one parameter subgroups (σg(t))t∈R.

Theorem 2.3.18. A tensor field on a smooth manifold X is invariant under a group oftransformations⇐⇒ its Lie derivative ≡ 0.

2.3.7. Adjoint representation and Maurer-Cartan form. Let G be a Lie groupof dimension p.

(1) A adjoint representation is a representation of G on g ∼= TeG.(2) Let

adg : G −→ G, h 7−→ adg(h) = Lg R−1g (h) = ghg−1

be the inner automorphism given in (2.3.31). Then we get

Teadg : TeG −→ TeG, v 7−→ (Teadg)v.

(3) Define

(2.3.35) Ad(g) : g −→ g, γ 7−→ Ad(g)(γ)

by requiring that the following diagram

gAd(g)−−−−→ g

∼=y x∼=

TeG −−−−→Teadg

TeG

γAd(g)−−−−→ Ad(g)(γ)y x

γe −−−−→ (Teadg)γe


is commutative. Thus

(Ad(g)(γ)) (h) = (TeLh)((Teadg)γe

)= (TeLh)

((Te(Lg R−1

g ))

γe

)= (TeLh)

(Tg−1 Lg

((TeR−1

g

)γe

))=

(Te

(Lh Lg R−1

g

))γe.

The adjoint representation of G on g:

Ad : G −→ L(g, g) := linear mappings on gg 7−→ Ad(g) : g −→ g(2.3.36)

γ 7−→ (Ad(g))(γ) =: g · γ · g−1.

(3.1) Ad is a representation:

((Ad(e))(γ)) (h) =(

Te

(Lh Le R−1

e

))γe = (TeLh)γe = γh

so Ad(e) = 1g. For g1, g2 ∈ G,

((Ad(g1) Ad(g2)) γ) (h) = ((Ad(g1)) ((Ad(g2)) (γ))) (h)

=(

Te

(Lh Lg1 R−1

g1

))((Ad(g2)) (γ)) (e)

=(

Te

(Lh Lg1 R−1

g1

)) ((Te

(Le Lg2 R−1

g2

))γe

)=

(Te

((Lh Lg1 R−1

g1

)(

Le Lg2 R−1g2

)))γe

=(

Te

(Lh Lg1g2 R−1

g1g2

))γe = (Ad(g1g2)γ) (h)

so Ad(g1) Ad(g2) = Ad(g1g2).(3.2) The center of G is the kernel Ker(g 7→ adg), which is Ker(Ad) if G is

connected.(4) The adjoint representation Ad of g on g:

(2.3.37) Ad := TeAd : TeG ∼= g −→ L(g, g).

For δ ∈ g (so δe ∈ TeG) we take g(t) the one parameter group satisfying

dg(t)dt

= δ(g(t)) with g(0) = e =⇒ δe =ddt

∣∣∣∣t=0

g(t).

ThenAd(δ) = (TeAd)(δe)

and

(Ad(δ))(γ) = ((TeAd) (δe)) (γ) =

((TeAd)

(ddt

∣∣∣∣t=0

g(t)))

(γ)

=ddt

∣∣∣∣t=0

((Ad(g(t))) (γ)) = [δ, γ].

♣ Exercise: Verify the last step.

2.3. LIE GROUPS 107

(5) Let (vα)1≤α≤p be a basis of g and [vα, vβ] = cγαβvγ =⇒

(Ad(δ)) (γ) = (Ad(δαvα)) (γβvβ) = δαγβ (Ad(vα)) (vβ)

= δαγβ[vα, vβ] = δαγβcγαβvγ.

Hence

(2.3.38) (Ad(δ))αβ = δγcα

γβ or Ad(δ) = δ · c.

(6) ∀ one parameter subgroup (g(t))t∈R of G =⇒ ∃ linear transformationA : g→ g such that

Ad(g(t)) = etA = exp(tA), A :=ddt

∣∣∣∣t=0

Ad(g(t)).

Let

v :=ddt

∣∣∣∣t=0

g(t) ∈ TeG ∼= g, δ• := (TeL•)v ∈ g.

We then get

Ad(δ) = (TeAd)(v) = (TeAd)(

ddt

∣∣∣∣t=0

g(t))=

ddt

∣∣∣∣t=0

Ad(g(t)) = A.

Since exp v = g(1), it follows that

Ad(exp δ) := Ad(exp v) = Ad(g(1)) = exp(A) = exp (Ad(δ)) = exp(δ · c).(7) The Killing form on g is the bilinear form

(2.3.39) B : g× g −→ R, (γ, δ) 7−→ B(γ, δ) := tr[Ad(γ), Ad(δ)].

(8) The canonical or Maurer-Cartan form ω on G is a one-form with valuesin g:

(2.3.40) ω(vg) :=(

TgL−1g

)vg ∈ TeG ∼= g, ω : TG → g.

Theorem 2.3.19. ω is left invariant and R∗gω = Ad(g−1) ω.

PROOF. For any g, h ∈ G we have

(L∗hω) (vh−1g) = ω((

Th−1gLh

)vh−1g

)=

(TgL−1

g

) ((Th−1gLh

)vh−1g

)=

(Th−1g

(L−1

g Lh

))vh−1g =

(Th−1gL−1

h−1g

)vh−1g = ω(vh−1g)

and

(R∗hω) (vgh−1) = ω((

Tgh−1 Rh

)vgh−1

)=

(TgL−1

g

) ((Tgh−1 Rh

)vgh−1

)=

(Tgh−1

(L−1

g Rh

))vgh−1 =

(Tgh−1

(L−1

h Rh L−1gh−1

))vgh−1

=(

Te

(L−1

h Rh

)) (ω(vgh−1)

)=

(Ad(h−1)

) (ω(vgh−1)

).

Thus R∗hω = Ad(h−1) ω.


2.4. Exterior differential forms

2.4.1. Exterior algebra. Let X be a Cr-manifold of dimension n, and r ≥ k +1 ≥ 1.

(1) A totally antisymmetric covariant p-tensor field is called a exterior dif-ferential p-form or form of degree p.

(2) Let

(2.4.1) Λp(X ) :=

p-forms of (class) Ck on X

.

Note that Λp(X ) ∈ Ck(X , T0,pX ).(2.1) ∀ α, β ∈ Λp(X ) =⇒ α + β ∈ Λp(X ).(2.2) ∀ α ∈ Λp(X ) and ∀ f ∈ Ck(X ) =⇒ f α ∈ Λp(X ).(2.3) Λp(X ) is a module over Ck(X ).(2.4) p > n =⇒ Λp(X ) = 0.

♣ Exercise: Verify (2.3) and (2.4).

(3) The exterior product or wedge product of a p-form and a q-form is amapping

(2.4.2) ∧ : Λp(X )×Λq(X ) −→ Λp+q(X ), (α, β) 7−→ α ∧ β

defined by

(α ∧ β)(v1, · · · , vp+q) :=1

p!q! ∑π∈Sp+q

sgn(π)

·α(vπ(1), · · · , vπ(p))β(πp+1, · · · , vπ(p+q)).(2.4.3)

In particular

(2.4.4) α ∧ β = α⊗ β− β⊗ α, ∀ α, β ∈ Λ1(X ).

Proposition 2.4.1. The wedge product satisfies(i) Associativity:

(α ∧ β) ∧ γ = α ∧ (β ∧ γ).(ii) Bilinearity:

α ∧ (β + γ) = α ∧ β + α ∧ γ,(α + β) ∧ γ = α ∧ γ + β ∧ γ,

f (α ∧ β) = f α ∧ β = α ∧ f β.

(iii) Non-commutativity:

(2.4.5) α ∧ β = (−1)pqβ ∧ α, ∀ α ∈ Λp(X ), ∀ β ∈ Λq(X ).

(iv) θ1, · · · , θp ∈ Λ1(X ) =⇒

(2.4.6) θ1 ∧ · · · ∧ θp = ϵ1···pi1···ip

θi1 ⊗ · · · ⊗ θip .

2.4. EXTERIOR DIFFERENTIAL FORMS 109

♣ Exercise: Prove Proposition 2.4.1.

(4) Let

(2.4.7) Λ(X ) :=⊕

0≤p≤nΛp(X ), Λx(X ) :=

⊕0≤p≤n

Λpx(X )

be the exterior algebra of X and of x, respectively.(4.1) f ∈ Λ0(X ) = Ck(X ) =⇒

(2.4.8) f ∧ α := f α.

(4.2) ∃ associative bilinear mapping

∧ : Λ(X )×Λ(X ) −→ Λ(X ).

(5) ∀ x ∈ X and take an arbitrary basis (θi)1≤i≤n of T∗xX .(5.1) A basis of Λ

px(X ) is the set of (n

p) independent p-forms:

(θi1 ∧ · · · ∧ θip)1≤i1<···<ip≤n or (θI)|I|=p with I = (1 ≤ i1 < · · · < ip ≤ n).

Here θI := θi1 ∧ · · · ∧ θip .

PROOF. ∀ α ∈ Λpx(X ) we have

α = αi1···ip θi1 ⊗ · · · ⊗ θip =1p!

ϵk1···kpi1···ip

αk1···kp θi1 ⊗ · · · ⊗ θip

=1p!

αk1···kp θk1 ∧ · · · ∧ θkp = ∑1≤i1<···<ip≤n

αi1···ip θi1 ∧ · · · ∧ θip

which can be written as ∑|I|=p αIθI or αIθI .

(5.2) ∀ x ∈ X and ∀ 0 ≤ p ≤ n

(2.4.9) dim Λpx(X ) =

(np

)=

n!p!(n− p)!

.

(5.3) A p-form with only one component α = aθi1 ∧ · · · ∧ θip is called amonomial.

(6) In the domain of a chart. Let (θi)1≤i≤n be an arbitrary basis for covariantvector fields in the domain of a chart. Then (θi)1≤i≤n is either the naturalbasis (dxi)1≤i≤n or a local frame (θi = ai

jdxj)1≤i≤n where [aij(x)]1≤i,j≤n

is an invertible differentiable matrix. Then (θI)|I|=p is a basis for the p-forms in the chosen chart.

(6.1) ∀ α = aiθi and β = βiθ

i ∈ Λ1(X ) =⇒

(2.4.10) α ∧ β = ∑1≤i,j≤n

αiβ jθi ∧ θ j = ∑

1≤i<j≤n(αiβ j − αjβi)θ

i ∧ θ j

so that (α ∧ β)ij = αiβ j − αjβi.(6.2) ∀ α ∈ Λp(X ) and β ∈ Λq(X ) =⇒

(2.4.11) α ∧ β = ∑|I|=p, |J|=q

αIθI ∧ βJθJ = ∑|I|=p, |J|=q

αI βJθI J , θI J := θI ∧ θJ


so that

(α ∧ β)i1···ip+q = ∑|I|=p, |J|=q

ϵI Ji1···ip+q

αI βJ .

(6.3) Let ω = α1 ∧ · · · ∧ αp with αi = αijθ

j ∈ Λ1(X ) =⇒

(2.4.12) ω = α1i1 · · · α

pip

θi1 ∧ · · · ∧ θip = ϵj1···jpI α1

j1 · · · αpjp

θI .

(7) Change of basis:θi 7−→ θi′ := ai′

j θ j.

For ω = ωIθI = ωI′θI′ ∈ Λp(X ) we have

ω =1p!

ωi′1···i′pθi′1 ∧ · · · ∧ θi′p = ωJθJ =

1p!

ωi′1···i′pai′1

j1· · · ai′p

jpθ j1 ∧ · · · ∧ θ jp

=1p!

ωi′1···i′pϵ

k1···kpJ ai′1

k1· · · ai′p

kpθJ = ωI′ϵ

k1···kpJ aI′

k1···kpθ J .

Therefore

(2.4.13) ωJ = ϵk1···kpJ aI′

k1···kpωI′ .

Observe that in aI′k1···kp

the index I ′ has the form I′ = (1 ≤ i′1 < · · · <i′p ≤ n), while k1, · · · , kp are any integers in 1, · · · , n.

(7.1) The terms

ϵk1···kpJ aI′

k1···kp

are the determinants of all the p× p matrices obtained from the n× nmatrix [ai′

j ]1≤i′,j≤n by eliminating n− p rows and n− p columns inall possible ways.

(7.2) If x 7→ x(x), then aj′

i = ∂xj′/∂xi and

(2.4.14) ωJ(x) = ωI′(x(x))D(xI′)

D(xJ).

2.4.2. Exterior differentiation. Let α ∈ Λp(X ) (of class Ck).(1) The exterior differentiation operator is

(2.4.15) d : Λp(X ) −→ Λp+1(X ), α 7−→ dα,

so that dα is of class Ck−1, and is called the exterior derivative or cobound-ary of α. satisfying

(1.1) d is linear:

d(α + β) = dα + dβ, d(λα) = λdα (λ ∈ R).

(1.2) ∀ α ∈ Λp(X ),

d(α ∧ β) = dα ∧ β + (−1)pα ∧ dβ.

(1.3) d2 = 0.(1.4) f ∈ Λ0(X ) =⇒ d f is the ordinary differential of f .(1.5) d is local: α = β on U ⊂ X open =⇒ dα = dβ on U .


Theorem 2.4.2. (1.1) – (1.4) uniquely determine the operator d.

PROOF. (Uniqueness) Let d be an operator which satisfies (1.1) – (1.4) and letα ∈ Λp(X ) with α = αIdxI . We shall show that dα is uniquely defined. By (1.2),

dα = d(

αIdxI)= dαI ∧ dxI + αI d(dxI).

By (1.4)dαI = dαI , dxi = dxi.

By (1.2) and (1.3),

d(

dxI)= d(dxI) = 0.

Hencedα = dαI ∧ dxI = dα

and d is the uniquely defined operator d.(Existence) For α = αIdxI define

(2.4.16) dα := dαI ∧ dxI = dαI ∧ dxI =∂αI

∂xk dxk ∧ dxI = ϵkIJ

∂αI

∂xk dxJ

where |J| = |I|+ 1 = p + 1. It is clear that dα satisfies (1.1) – (1.5).Moreover, this definition does not depend on the choice of coordinates. Con-

sider the change of coordinates xi 7→ xi′ and obtain

(dα)x = d(

1p!

αi1···ip dxi1 ∧ · · · ∧ dxip

)=

1p!

d(

αi′1···i′p(x(x))dxi′1 ∧ · · · ∧ dxi′p

)=

1p!

∂αi′1···i′p∂xk′

∂xk′

∂xj dxj ∧ dxi′1 ∧ · · · ∧ xi′p

=1p!

∂αi′1···i′p∂xk′ dxk′ ∧ dxi′1 ∧ · · · ∧ dxi′p = (dα)x.

Finally, we show that dα is a (p + 1)-form defined on X . Define

di := dUi .

By (1.5) we have di = dj on Ui ∩ Uj so putting together gives the existence of d onX .

dα = 0 for ∀ α ∈ Λn(X ). In our notion, we see that ∂αI/∂xk are not the com-ponents of a (p + 1)-tensor, but ϵkI

J ∂αI/∂xk are.

Let A• = (Ap)p be a graded algebra.(1) A linear mapping

T : A• −→ A•

is a differential operator of degree s on A• if

T2 = 0 and T : Ap −→ Ap+s.


(2) (Λ(X ), d) is a graded algebra with differential operator of degree 1 or adifferential graded algebra (DGA).

Example 2.4.3. (1) f ∈ Λ0(R3) =⇒

d f = d f =∂ f∂xi dxi.

(2) α = Adx + Bdy + Cdz ∈ Λ1(R3) =⇒

dα =

(∂C∂y− ∂B

∂z

)dy ∧ dz +

(∂A∂z− ∂C

∂x

)dz ∧ dx +

(∂B∂x− ∂A

∂y

)dx ∧ dy.

(3) For ω = Pdy ∧ dz + Qdz ∧ dx + Rdx ∧ dy ∈ Λ2(R3) =⇒

dω =

(∂P∂x

+∂Q∂y

+∂R∂z

)dx ∧ dy ∧ dz.

2.4.3. Reciprocal image of a form (pull-back). Let f : X → Y be a differen-tiable mapping and let v ∈ TxX . Then Txf(v) ∈ Tf(x)Y . Define

(2.4.17) f∗ : Λp(Y) −→ Λp(X ), ω 7−→ f∗ω.

(1) The reciprocal image f∗ω of ω ∈ Λp(Y) is defined by

(2.4.18) (f∗ω)x(v1, · · · , vp) := ωf(x)(f′x(v1), · · · , f′x(vp)).

(2) f∗ω is also called the form induced by f from ω.(3) Locally

(f∗ω)i1···ip =∂yα1

∂xi1· · · ∂yαp

∂xipωα1···αp =

D(yI)

D(xi1 , · · · , xip)ωI

and

(2.4.19) f∗ω = (f∗ω)I(x)dxI .

Theorem 2.4.4. The induced mapping f∗ (2.4.17) is a homomorphism on Λ(Y).

PROOF. (i) f∗ is a homomorphism on the vector space Λp(Y):f∗(λω + µθ) = λf∗ω + µf∗θ.

(ii) f∗ is a homomorphism on the graded algebra Λ(Y):f∗(ω ∧ θ) = f∗ω ∧ f∗θ.

(iii) f∗ is a homomorphism on the DGA:

f∗(dω) = d(f∗ω) ⇐⇒ f∗ d = d f∗.

Indeed,

differential form = ∑finite sum

gi ∧ dh(1)j ∧ · · · ∧ dh(j)j


Figure 2.18: Reciprocal image of ω

where all gi and h(•)j are functions. We shall check

f∗(dg) = d(g f) = d(f∗g), f∗(ddg) = 0 = dd(f∗g) = df∗(dg).

Hence f∗ is a homomorphism on Λ(Y).

Example 2.4.5. (1) If f is a diffeomorphism, then∫f()

ω =∫

f∗ω, ∀ ω ∈ Λ(f()).

(2) Let :=

(φ, θ) ∈ R2 : 0 < φ < 2π, 0 < θ < π

and

g : −→ S2 \ set of measure 0, (θ, φ) 7−→ (sin θ cos φ, sin θ sin φ, cos θ),

and the inclusion

ι : S2 −→ R3, (sin θ cos φ, sin θ sin φ, cos θ) 7−→ (x1, x2, x3).

Let ω = Pdx2 ∧ dx3 + Qdx3 ∧ dx1 + Rdx1 ∧ dx2. Then

(ι g)∗ω = P(cos θ sin φdθ + sin θ cos φdφ) ∧ (− sin θ dθ)

+ Q(− sin θ dθ) ∧ (cos θ cos φdθ − sin θ sin φdφ)

+ R(cos θ cos φdθ − sin θ sin φdφ) ∧ (cos θ sin φdθ + sin θ cos φdφ)

= (P sin θ cos φ + Q sin θ sin φ + R cos θ) sin θ dθ ∧ dφ.


Figure 2.18: Example 2.4.5 (2)

Therefore∫ 2π

0dφ∫ π

0dθ (P sin θ cos φ + Q sin θ sin φ + R cos θ) sin θ =

∫(ι g)∗ω

=∫(ιg)()

ω =∫

S3Pdx2 ∧ dx3 + Qdx3 ∧ dx1 + Rdx1 ∧ dx2.

2.4.4. Derivation and antiderivation on Λ(X ). Let T be a linear operator onΛ(X ):

T(λω + µθ) = λT(ω) + µT(θ).

(1) T is of degree p if

T : Λr(X ) −→ Λr+p(X )

for all r. In this case we write deg(T) = p.(2) T is a derivation on Λ(X ) if deg(T) is even and if it obeys Leibniz rule

(2.4.20) T(ω ∧ θ) = T(ω) ∧ θ + ω ∧ T(θ).

(3) T is an anti-derivation on Λ(X ) if deg(T) is odd and if it obeys anti-Leibniz rule

(2.4.21) T(ω ∧ θ) = T(ω) ∧ θ + (−1)deg(ω)ω ∧ T(θ).

(4) T is local on Λ(X ) if ∀ U ⊂ X open, T(ω)|U depends only on ω|U .


Proposition 2.4.6. (1) The commutator of two derivations is a derivation. The anti-commutator of two anti-derivations is a derivation. The commutator if a derivation withan anti-derivation is an anti-derivation.

(2) Two derivations (anti-derivations) are equal if they are equal on 0-forms and on1-forms.

(3) If T is a local derivation or anti-derivation which commutes with the exteriorderivative d, then it is fully determined by its action on 0-forms.

PROOF. We only prove (3). In fact T(gd f ) = T(g) ∧ d f ± g T(d f ) = T(g) ∧d f ± gd(T f ).

♣ Exercise: Complete the proof of Proposition 2.4.6.

The Lie derivative is a derivation on Λ(X ):

Lv(s⊗ u) := Lvs⊗ u + s⊗Lvu,(2.4.22)Lv(ω ∧ θ) = Lvω ∧ θ + ω ∧Lvθ,(2.4.23)

Lvω =1p!

(vk∂kωi1···ip + pωki2···ip ∂i1 vk

)dxi1 ∧ · · · ∧ dxip .(2.4.24)

The interior product or inner product of a form ω and a vector v is denotedιvω or vyω.

(1) ιv is an anti-derivation.(2) ιv f = 0 for all f ∈ Λ0(X ).(3) ιvdxi = vi.(4) ∀ ω ∈ Λo(X ):

(2.4.25) ιvω =1

(p− 1)!vjωji2···ip dxi2 ∧ · · · ∧ dxip = vjωjIdxI

with |I| = p− 1.

Proposition 2.4.7. We have(i) ι2v = 0.

(ii) Lv = d ιv + ιv d.(iii) [Lv, ιw] := Lv ιw − ιw Lv = ι[v,w].(iv) dθ(v, w) = Lv(θ(w)) − Lw(θ(v)) − θ([v, w]). More generally, ∀ ω ∈

Λp(X ),

dω(v0, v1, · · · , vp) = ∑0≤i≤p

(−1)ivi(ω(v0, · · · , vi, · · · , vp)

)+ ∑

0≤i<j≤p(−1)i+jω([vi, vj], v0, · · · , vi, · · · , vj, · · · , vp).(2.4.26)

(v) L[v,w] = [Lv, Lw].


PROOF. (i) Clearly.(ii) By Proposition 2.4.6 and

Lv f = v f = ιvd f = (ιv d + d ιv) f ,d Lv = Lv d,

d (ιv d + d ιv) = (d ιv + ιv d) d

we see that Lv = ιv d + d ιv.(iii) ∀ f ∈ Λ0(X ) and α = αidxi ∈ Λ1(X ) =⇒

[Lv, ιw] f = 0, ι[v,w] f = 0,

[Lv, ιw]α = ∂ℓ(wiαi)vℓ − wivℓ∂ℓαi − wℓαi∂ℓvi = ι[v,w]α.

(iv) It follows from

dθ(v, w) = vjwidθ

(∂

∂xj ,∂

∂xi

)= (∂jθi − ∂iθj)vjwi,

Lv(θ(w)) = vj∂j(θiwi),

θ([v, w]) = θi(vj∂jwi − wj∂jvi) = ι[v,w]θ.

(v) From (iv), we have

ιwιvdθ = Lvιwθ −Lwιvθ − ι[v,w]θ.

Taking θ = d f yieldsL[v,w] f = LvLw f −LwLv f .

By Proposition 2.4.6, L[v,w] = [Lv, Lw].

2.4.5. Forms defined on a Lie group. Let G be a Lie group of dimension n.(1) ω ∈ Λp(G) is left invariant if

(2.4.27) L∗gω = ω, ∀ g ∈ G.

(2) ω is left invariant⇐⇒ ωe = (L∗gω)e = ωg TeLg.(3) We have

(2.4.28) dim (left-invariant p-forms) = dim Λpe (G) =

(np

).

♣ Exercise : Verify (2.4.28).

(4) ω ∈ Λp(G) is right invariant if

(2.4.29) R∗gω = ω, ∀ g ∈ G.

(5) ω is a left (resp. right) invariant form⇐⇒ dω is left (resp. right) invari-ant:

L∗gdω = dL∗gω = dω.

(6) Let

(2.4.30) g∗ := dual of g = left invariant 1-forms on G

with basis (vα)1≤α≤n for g and dual basis (θβ)1≤β≤n.


(6.1) Since dθα is left invariant, it follows that

(2.4.31) dθα := − ∑1≤B<Γ≤n

cαBΓθB ∧ θΓ = −1

2cα

βγθβ ∧ θγ

which is the Maurer-Cartan structure equations.(6.2) The strict components of dθα are, up to −1, the structure constants

of the group [vβ, vγ] = cαβγvα.

PROOF. Indeed,

(dθα)βγ = dθα(vβ, vγ) = Lvβ(θα(vγ))−Lvγ(θ

α(vβ))− θα([vβ, vγ])

= −θα([vβ, vγ]) = −θα(

cδβγvδ

)= −cα

βγ

Thus dθα = −∑1≤β<γ≤n cαβγθβ ∧ θγ.

Theorem 2.4.8. (Theorem 2.3.11) A Lie group G has vanishing structure constants⇐⇒ G is locally isometric to Rn.

PROOF. It remains to prove that (cαβγ = 0 =⇒ G is locally isometric to Rn).

Since cαβγ = 0, dθα = 0. Poincare lemma =⇒ in the domain of any local chart ∃ n

independent differentiable functions φα such that

θα = dφα, 1 ≤ α ≤ n.

Since θα is left invariant, it follows that

(L∗gθα)h = θαh = (dφα)h = hα,

where we use (φα)1≤α≤n as the coordinate functions of U . But

(L∗gθα)h = (L∗gdφα)h = (dL∗g φα)h = d(φα Lg)h = dLα(gγ, hβ) =∂Lα

∂hβdhβ,

where Lα(gγ, hβ) are the coordinates of gh. Hence

∂Lα

∂hβ= δα

β, 1 ≤ α, β ≤ n.

Therefore

Lα(gγ, hβ) = Φα(gγ) + hα.

WLOG, we may assume that e ∈ U . Since hα = 0 for h = e, it follows that

Φα(gγ) = Lα(gγ, 0) = (ge)α = gα

and then Lα = gα + hα.


2.4.6. Vector-valued differential forms. Let X be a smooth manifold of di-mension n.

(1) An (exterior) p-form φ at x which values in a given vector space V is atotally anti-symmetric p-linear map from TxX to V:

φx : TxX × · · · × TxX︸︷︷︸p

−→ V, (v1, · · · , vp) 7−→ φx(v1, · · · , vp).

We write it as φx ∈ Λpx(X )⊗V and φ ∈ Λp(X )⊗V‘.

(1.1) If V is a r-dimensional real vector space, then

φx(v1, · · · , vp) = φαx(v1, · · · , vp)eα

where (eα)1≤α≤r is a base of V, φαx(v1, · · · , vp) ∈ R, and (v1, · · · , vp)

7→ φαx(v1, · · · , vp) is p-linear, totally anti-symmetric. Hence φα

x ∈Λ

px(X ) and

φx = φαx ⊗ eα.

(1.2) An (exterior, differential) p-form φ with values in a given finitedimensional real vector space V on X is an assignment

X −→ ΛpxX ⊗V, x 7−→ φx = φα

x ⊗ eα.

Thenφ = φα ⊗ eα, φα ∈ Λp(X ).

(1.3) φ is of class Ck on X if φα are of class Ck.(2) The exterior differential of φ ∈ Λp(X )⊗V is defined by

(2.4.32) dφ = d(φα ⊗ eα) := dφα ⊗ eα.

(2.1) dφ does not depend on the choice of a basis in V,

eα = Aα′α eα′ , φ = φα ⊗ eα = φα′ ⊗ eα′ , φα′ = Aα′

α φα.

Then

dφα′ ⊗ eα′ = d(

Aα′α φα

)⊗ eα′ = Aα′

α dφα ⊗ eα′ = dφα ⊗ eα.

(2.2) Basic properties:

d(φ+ ψ) = dφ+ dψ,

d2φ = 0,d⟨a,φ⟩ = ⟨a, dφ⟩

where a ∈ V∗ and ⟨a,φ⟩x(v1, · · · , vp) = ⟨a,φx(v1, · · · , vp)⟩.(3) Assume that the vector space V is endowed with a Lie algebra structure

by a bracket [eα, eβ]. Define the bracket of a p-form and a q-form on Xwith values in V by

[φ, ψ] := (φα ∧ ψβ)⊗ [eα, eβ] ∈ Λp+q(X )⊗V.

(3.1) [φ, ψ] does not depend on the choice of a basis in V:

(φα′ ∧ ψβ′)⊗ [eα′ , eβ′ ] = (Aα′α Aβ′

β φα ∧ ψβ)⊗ [eα′ , eβ′ ] = (φα ∧ ψβ)⊗ [eα, eβ].

2.5. INTEGRATION 119

(3.2) Basic properties:

[φ1 +φ2, ψ] = [φ1, ψ] + [φ2, ψ],(2.4.33)

[φ, ψ] = (−1)1+|φ||ψ|[ψ,φ],(2.4.34)

0 = (−1)|φ||θ|[φ, [ψ, θ]] + (−2)|ψ||φ|[ψ, [θ,φ]]

+(−1)|θ||ψ|[θ, [φ, ψ]],(2.4.35)

d[φ, ψ] = [dφ, ψ] + (−1)|φ|[φ, dψ].(2.4.36)

♣ Exercise : Verify (2.4.33) – (2.4.36).

2.5. Integration

Two coordinate systems (xi)1≤i≤n and (yj)1≤j≤n on an open set of Rn are saidto define the same orientation if

(2.5.1)D(xi)

D(yj)> 0

at all points. It defines an equivalence relation on all coordinate systems.(1) A differentiable manifold X is said to be orientable if ∃ an atlas such that

on the overlap U ∩V of any two charts (U ,φ) and (V , ψ), D(φi)/D((ψj) >0.

(2) A manifold is oriented it is orientable in terms of such atlas in (1).(3) An orientation at x ∈ X can also be defined in terms of the orientation of

TxX .(4) A connected differentiable manifold is orientable ⇐⇒ its frame bundle

consists of two disjoint sets.(5) If L is an n-dimensional real vector space, then Λn(L) ∼= R so that we can

give L a natural orientation.

Over non-orientable manifolds, de Rham developed a theory of integration.

2.5.1. Integration. We consider an integration of a differential n-form on ann-dimensional paracompact oriented manifold X .

(1) A subset Y ⊂ X is of (Lebesgue) measure zero if it is a countable unionof images in X , under inverse coordinate mappings, of sets of measurezero in Rn.

(2) A mapping f : X → Z is defined almost everywhere (a.e.) if it is definedexcept on a subset of measure zero.

(3) Integration of an n-form with compact support in the domain U of achart with coordinates (x1, · · · , xn).

(3.1) Let ω be an n-form vanishing outside a compact set contained in U .(3.2) ω is integrable onX if its component ω1···n is integrable on Rn. Then

define

(2.5.2)∫X

ω ≡∫U

ω :=∫

Rnω1···n(x1, · · · , xn)dx1 · · · dxn.


(3.3) The definition does not depend on the choice of coordinates in U ,provided that they are compatible with the orientation:

(xi)1≤i≤n 7−→ (xi)1≤i≤n,D(xi)

D(xj)> 0.

Then

ω1···n(xj) = ω1···n(xi)D(xℓ)D(xk)

and∫Rn

ω1···n(xj)dx1 · · · dxn =∫

Rnω1···n(xi)

D(xℓ)D(xk)

dx1 · · · dxn

=∫

Rnω1···n(xi)dx1 · · · dxn.

(4) Integration of an n-form with arbitrary support in X .(4.1) A partition of unity on X is a collection of functions θk ≥ 0 on X

with the following properties:(4.1.1) (supp(θk))k∈I is locally finite (i.e., ∀ x ∈ X ∃ finite number of

functions θk = 0);(4.1.2) supp(θk) is compact;(4.1.3) ∑k∈I θk ≡ 1 on X .

(4.2) The partition is of class Cr if the θk’s are of class Cr.(4.3) A partition of unity (θk))k∈I is subordinate to a covering (Uk)k∈I of

X if supp(θk) ⊂ Uk for all k ∈ I.(4.4) According to Theorem 2.5.1, we choose a partition (θk)k∈I of unity

that subordinates to an atlas of X . For an n-form ω, define

(2.5.3)∫X

ω := ∑k∈I

∫X

θkω

whenever it converges.(4.5) ∀ continuous b-form is integrable on a compact n-manifold.(4.6) Y ⊂ X and ω ∈ Λn(X ) =⇒

(2.5.4)∫Y

ω =∫X

χYω

where χY is the characteristic function of Y .

Theorem 2.5.1. If X is paracompact, then ∃ partition of unity subordinates to any pre-signed locally finite covering. If, moreover, X is a Cr-manifold, this partition of unity canbe required to be Cr.

We also use the notion:

(2.5.5) ⟨X , ω⟩ :=∫X

ω.


Figure 2.19: 0-simplex, 1-simplex, and 2-simplex

Proposition 2.5.2. (1) We have∫X(λω + µθ) = λ

∫X

ω + µ∫X

θ,∫X

ω =∫X1

ω +∫X2

ω

with X = X1 ∪ X2.(2) The integral of an n-form ω on A ⊂ X does not change if A is modified by a set

of measure zero.(3)If f : X → Y is an orientation-preserving diffeomorphism, ω ∈ Λn(Y , then

(2.5.6)∫Y

ω =∫

f(X )ω =

∫X

f∗ω.

2.5.2. Stokes’ theorem. We first state Stokes’ theorem:

(2.5.7)∫C

dω =∫

∂Cω or ⟨C, dω⟩ = ⟨∂C, ω⟩.

Here all forms are at least C1.(1) A p-simplex is a subset of Rn defined in terms of p + 1 linearly indepen-

dent points (see Figure 2.19).

(2) A rectangle P is a naturally oriented subset of Rp defined by

(2.5.8) P :=(x1, · · · , xp) ∈ Rp|ai ≤ xi ≤ bi, 1 ≤ i ≤ p

.

(3) Simplexes ≈ Rectangles: A simplex can be decomposed into sets diffeo-morphic to rectangles and vice-versa. But

simplexes homology, rectangles integration.

(4) We have

rectangle −→

elementary p-chain −−−−→ p-chainyelementary domain of integration −−−−→ domain of integration

(5) An elementary p-chain c on X is a pair (P , f), where P is a p-rectanglein Rp and f : U → X , with P ⊂ U ⊂ Rp, is a differentiable mapping.

(5.1) Im(f|P ) = supp(c).(5.2) c = (P , f) is an elementary p-domain of integration if f is a diffeo-

morphism of U onto a differentiable p-submanifold of X . Then c orsupp(c) is also referred to as an elementary p-domain of integration.


(5.3) ∀ ω ∈ Λp(X ) and ∀ elementary p-domain c = (P , f) of integration:

(2.5.9)∫

supp(c)ω =

∫P∗ω.

(6) A p-chain C on X is a formal linear combination of elementary p-chainswith real coefficients:

(2.5.10) C := ∑i

λici.

(7) A formal locally finite linear combination, with coefficients ±1, of ele-mentary domains of integration is called a domain of integration.

Integrals of p-forms on p-chains.

(1) ω ∈ Λp(X ) and elementary p-chain c = (P , f) =⇒

(2.5.11)∫

cω :=

∫P

f∗ω.

(2) ω ∈ Λp(X ) and p-chain C = ∑i λici =⇒

(2.5.12)∫C

ω := ∑i

λi

∫ci

ω =: ⟨C, ω⟩.

(2.1) ω is continuous and C is finite: (2.5.12) is well-defined.(2.2) ω is continuous with compact support and C is locally finite: (2.5.12)

is well-defined.(2.3) ⟨C, ω⟩ is bilinear in C and ω.

(3) Two chains C and C ′ are equal if∫C

ω =∫C ′

ω

for all ω.

Boundaries.

(1) The boundary ∂P of a rectangle P in Rp is the 2p rectangles in Rp−1

defined by the faces xi = ai and xi = bi of the rectangle. The p − 1coordinates of a point on these faces are (x1, · · · , xi, · · · , xp) with aj ≤xj ≤ bj (1 ≤ j ≤ p, j = i).

(2) The boundary ∂P is said to be coherently oriented withP when the facesare given the following orientations:

(x1, · · · , xi, · · · , xp) for

xi = ai and i even,xi = bi and i odd,

and

the opposite orientation for

xi = ai and i odd,xi = bi and i even.


(3) Let ∂P be the coherently oriented boundary of a rectangleP in Rp, whereboth P and Rp have the natural orientation.

Let (e1, · · · , ep−1) be a frame, with orientation compatible with thatof ∂P , at a point x in a face of P . If (e1, · · · , ep−1, ep) is a frame at x, withorientation compatible with that of Rp, then ep is inside P for p even andoutside P for p odd (See Figure 2.21).

(4) C = ∑i λici =⇒∂C := ∑

iλi∂ci.

(5) Manifold X with boundary (See Figure 2.22):

∂X :=

x ∈ X : φ(x) ∈ ∂Rn−1

that is a (n− 1)-dimensional manifold.

X is of class Cr =⇒ ∂X is of class Cr.

Observe that ∂(∂X ) = ∅.(6) An oriented n-dimensional manifoldX is triangulable if it can be decom-

posed into a union of adjacement n-dimensional elementary domains ofintegration with orientation compatible with that of X .

(6.1) Compact oriented manifolds are triangulable.(6.2) C is the domain of integration that is equal to the sum of the elemen-

tary domains of integration which triangulate X =⇒

(2.5.13)∫X

ω :=∫C

ω, ∀ ω ∈ Λn(X ).

Mapping of chains.

(1) g : X → Y differentiable mapping, c = (P , f) elementary p-chain on X=⇒ g(c) = (P , g f), ∀ ω ∈ Λp(Y)

(2.5.14)∫

g(c)ω =

∫P(g f)∗ω =

∫P

f∗g∗ω =∫

cg∗ω.

For C := ∑i λici, set

(2.5.15) g(C) := ∑i

λig(ci).

(2) g is called proper if ∀ compact subset V ⊂ Y , g−1(V) is compact in X .(2.1) g is proper, C is locally finite chain =⇒ g(C) is locally finite.(2.2) g is proper, ω is compact support =⇒ g∗ω is compact support.

Theorem 2.5.3. If g : X → Y is proper differentiable mapping, ω ∈ Λp(Y) has compactsupport, and C is locally finite or finite, then

(2.5.16)∫

g(C)ω =

∫C

g∗ω.


Figure 2.20: Coherented orientation

(3) g : S1 → S1, differentiable two-one mapping, ω ∈ Λ1(S1). S1 can betriangulated by two elementary domains of integration ci = (P , fi) (SeeFigure 2.23):

S1 = supp(c1) ∪ supp(c2).

Then∫S1

g∗ω =∫

c1∪c2

g∗ω =∫

g(c1+c2)ω = 2

∫S1

ω =∫

S1ω =

∫g(S1)

ω,

because g(c1 + c2) is NOT a triangulation of S1.

Theorem 2.5.4. If X ,Y are connected and oriented, and f : X → Y is an orientation-preserving proper differentiable mapping, then ∃ integer deg(f), called the degree of f,such that

(2.5.17)∫X

f∗ω = deg(f)∫Y

ω

for ∀ n-form ω on Y with compact support.

Proof of (2.5.7). We give a proof to the particular case of a 2-rectangle (SeeFigure 2.24). Let

ω = a(x, y)dx + b(x, y)dy.


Figure 2.21: Orientation of boundaries

Figure 2.22: Manifolds with boundaries


Figure 2.23: Example

Thendω = (bx − ay)dx ∧ dy.

By integration by parts we obtain∫P

dω =∫P

(bx − ay

)dxdy

=∫ D

C[b(B, y)− b(A, y)] dy−

∫ B

A[a(x, D)− a(x, C)] dx

=∫ B

Aa(x, C)dx +

∫ D

Cb(B, y)dy +

∫ A

Ba(x, D)dx +

∫ C

Db(A, y)dy

=∫

∂Pω.

2.5.3. de Rham and Poincare theorems. BY Stokes’ theorem,

⟨∂2C, ω⟩ = ⟨∂C, dω⟩ = ⟨C, d2ω⟩ = 0.

Hence

(2.5.18) d2 = 0 =⇒ ∂2 = 0.

(1) Homology and cohomology:

Λ(X ) := forms, C(X ) := finite chains.(1.1) Λ(X ) (resp. C(X )) is a collection of vector spaces Λp(X ) (resp.

Cp(X )) over R.


Figure 2.24: Example

(1.2) Λ(X ) (resp. C(X )) has a coboundary (resp. boundary) operatorsuch that

d : Λ(X ) −→ Λ(X ), dΛp(X ) ⊂ Λp+1(X ), d2 = 0,

∂ : C(X ) −→ C(X ), ∂Cp(X ) ⊂ Cp−1(X ), ∂2 = 0.

(1.3) A form ω such that dω = 0 is called a cocycle or closed form.(1.4) A finite chain C such that ∂C = 0 is called a cycle.(1.5) A form ω such that ω = dθ is called a coboundary or exact form.(1.6) A finite chain C such that C = ∂B is called a boundary.(1.7) Let

Zp(X ) := cocycles of degree p,Zp(X ) := cycles of degree p,Bp(X ) := coboundaries of degree p,Bp(X ) := boundaries of degree p.

Then Bp(X ) ⊂ Zp(X ( and Bp(X ) ⊂ Zp(X ).(1.8) Let

Hp(X ) :=Zp(X )

Bp(X ), p-cohomology vector space,(2.5.19)

Hp(X ) :=Zp(X )

Bp(X ), p-homology vector space.(2.5.20)

We also call (2.5.19) the de Rham cohomology, and the (total) deRham cohomology is defined by

(2.5.21) H•(X ) :=⊕

0≤p≤nHp(X ).

We say

ω1 ∼ ω2 (cohomologous) ⇐⇒ ω1 −ω2 = dθ,

C1 ∼ C2 (homologous) ⇐⇒ C1 − C2 = ∂B.

Then we have cohomology class [ω] and homology class [C].


♣ Exercise : Define [ω1] · [ω2] := [ω1 ∧ ω2]. Show that the product is inde-pendent of the choice of ω1 and ω2.

(2) Convention for 0-forms and 0-chains:(2.1) B0(X ) = ∅,(2.2) Z0(X ) = f ∈ Λ0(X ) : d f = 0 =⇒

(2.5.22) H0(X ) = #connected components of X .

(3) The Euler-Poincare characteristic of X :

χ(X ) := ∑0≤i≤n

(−1)ibi(X ),(2.5.23)

bi(X ) := dim Hi(X ),(2.5.24)

bi(X ) := dim Hi(X ).(2.5.25)

Theorem 2.5.5. (Poincare Lemma) If an open subset U ⊂ X is diffeomorphic to Rn,then all closed forms on U of degree p ≥ 1 are exact (i.e., bp(U ) = 0, ∀ p ≥ 1).

PROOF. Let ω ∈ Λp(U ) with dω = 0. Define a linear operator

T : Λp(Rn) −→ Λp−1(Rn)

satisfyingT d + d T = 1.

· · · // Λp−1(Rn)d //

1

Λp(Rn)d //

1

T

yyrrrrrrrrrrΛp+1(Rn)

1

//

T

yyrrrrrrrrrr· · ·

· · · // Λp−1(Rn)d

// Λp(Rn)d

// Λp+1(Rn) // · · ·

Define

(Tω)x :=∫ 1

0tp−1(ιxω)txdt, x ∈ Rn.

Then

[(T d + d T)ω]x =∫ 1

0tp−1 [(ιx d + d ιx)ω]tx dt

=∫ 1

0tp−1(Lxω)txdt =

∫ 1

0

ddt

(tpωtx) dt = ωx,

where we usedddt

(tpωtx) = ptp−1ωtx + tp−1 ∂ω

∂xi xi = tp−1(Lxω)tx.

Hence ω = d(Tω) + T(dω) = d(Tω).

♣ Exercise : H1(R1) = R.


(4) Theorem 2.5.5 does NOT apply if the form ω fails to be differentiable atcertain points of Rn:

ω =−ydx + xdy

x2 + y2 = d(

tan−1 yx

)on R2 \ (0, 0).

(5) Compact cohomology:

Hqc (X ) :=

Zqc (X )

Bqc (X )

,(2.5.26)

bqc (X ) := dim Hq

c (X ).(2.5.27)

Here, replacing Λq(X ) by Λqc(X ), the set of C∞ exterior q-forms with

compact support, in the definition of Zq(X ), Bq(X ), and Hq(X ), we ob-tain Zq

c (X ), Bqc (X ), and Hq

c (X ).

♣ Exercise : H1c (R

1) = R.

Theorem 2.5.6. (Poincare Lemma: compact support) In Rn, a closed p-form withcompact support is the coboundary of a (p− 1)-form with compact support if p ≤ n− 1,and an n-form ω with compact support is the coboundary of an (n− 1)-form with compactsupport if and only if

∫Rn ω = 0. Consequently

(2.5.28) bqc (R

n) =

0, 0 ≤ q ≤ n− 1,1, q = n.

(6) X compact =⇒ Λ(X ) = Λc(X ).

b0 b1 · · · bp · · · bn−1 bn remarksΛ(Rn) 1 0 0 0 · · · 0 0Λc(Rn) 0 0 0 0 · · · 0 1 bp = bn−p

cΛ(Sn), Λc(Sn) 1 0 0 0 · · · 0 1Λ(Tn), Λc(Tn) 1 n · · · (n

p) · · · n 1 bp = bpc = bn−p

(7) Duality:

⟨C, ω⟩ =∫C

ω, ⟨chain, form⟩ = number.

(7.1) Finite chains, forms with arbitrary support:

(2.5.29) C(X )×Λ(X ) −→ R, (C, ω) 7−→ ⟨C, ω⟩.(7.2) Infinite chains, forms with compact support: ω ∈ Zp(X ) and C ∈

Zp(X ) =⇒

period of ω :=∫C

ω = ⟨C, ω⟩ = ⟨[C], [ω]⟩.


Then(7.2.1) A closed form is exact if and only if all its periods vanish.(7.2.2) (de Rham theorem):

H∞p (X )× Hp

c (X )→ R is non-degenerate.

Here H∞p (X ) is the infinite chain homology.

(7.2.3) X is compact and oriented, ω ∈ Λn(X ) = 0 =⇒ (⟨X , ω⟩ = 0⇐⇒ ω = dθ). Indeed,

ω = dθ ⇐⇒ ⟨C, ω⟩ = 0, ∀ C ∈ Zn(X )

⇐⇒ ⟨λX + ∂B, ω⟩ = 0, ∀ λ ∈ R

⇐⇒ ⟨X , ω⟩ = 0.

Since dim Hn(X ) = 1, it follows that any element C in Zn(X )can be written as λX + ∂B.

Theorem 2.5.7. (de Rham theorem) The mapping

(2.5.30) Hp(X )× Hp(X ) −→ R, ([C], [ω]) 7−→ ⟨C, ω⟩is a bilinear non-degenerate mapping which establishes the duality of Hp(X ) and Hp(X )and the equality

bp(X ) = bp(X )

when Hp(X ) (Hp(X )) is finite-dimensional.

Theorem 2.5.8. (Poincar’e duality theorem) X is oriented and n-dimensional =⇒

(2.5.31) Hp(X ) ∼=(

Hn−pc (X )

)∗and then bp(X ) = bn−p

c (X ).

2.6. Exercises and problems

CHAPTER 3

Riemannian manifolds

3.1. Pseudo-Riemannian structures

Let X be a smooth manifold of dimension n.

3.1.1. Riemannian and Lorentz manifolds. A pseudo-Riemannian manifoldis a manifold X together with a continuous 2-covariant tensor field g, called themetric tensor, such that

(i) g is symmetric,(ii) ∀ x ∈ X , the bilinear form gx is non-degenerate (i.e., gx(v, w) = 0 for all

v ∈ TxX , then w = 0).(1) A pseudo-Riemannian manifold is Riemannian or definite or proper if

gx(v, v) > 0, ∀ 0 = v ∈ TxX , x ∈ X .

Otherwise, X is called indefinite or improper.(2) In terms of a moving frame (θi)1≤i≤n, we have

(3.1.1) g = gij θi ⊗ θ j or ds2 = gijθiθ j,

where θiθ j = 12 (θ

i ⊗ θ j + θ j ⊗ θi).(3) The inner product or scalar product on TxX :

(3.1.2) (v|w) := gx(v, w), ∀ v, w ∈ TxX .

If (ei)1≤i≤n is the basis dual to (θi)1≤i≤n, then

(3.1.3) gij := (ei, ej), 1 ≤ i, j ≤ n.

(4) The norm ||v|| of v ∈ TxX is defined by

(3.1.4) ||v||2 := gx(v, v) = gijvivj.

(4.1) v ∈ TxX is called a null vector if ||v|| = 0.(4.2) The null cone or light cone at x:

(3.1.5) v ∈ TxX : ||v|| = 0.(5) Fix u ∈ TxX , define a mapping

(u|•) : TxX −→ R, v 7−→ (u|v).The canonical isomorphism is

(3.1.6) TxX −→ T∗xX , u 7−→ (u|•) := u∗ = gijuiθ j = ujθj.

We call ui the contravariant components of u and uj the covariant com-ponents of u.

131

132 3. RIEMANNIAN MANIFOLDS

(5.1) An inner product is defined on T∗xX by

(3.1.7) (u∗|v∗) := (u|v) = gijuivj = gijuivj.

(5.2) Similarly, ∃ canonical isomorphisms between

⊗pTxX , ⊗pT∗xX , ⊗qTxX ⊗⊗p−qT∗xX .

(6) ∃ basis (ei′)1≤i′≤n for TxX such that

(3.1.8) gx(v, v) = gijvivj = gi′ j′vi′vj′ = ∑

1≤i′≤k(vi′)2 − ∑

k+1≤i′≤n(vi′)2.

(6.1) The index of gx is k (does NOT depend on (ei′)1≤i′≤n).(6.2) The signature of gx is k− (n− k) = 2k− n.(6.3) In terms of the basis (θi′)1≤i′≤n dual to (ei′)1≤i′≤n:

(3.1.9) gx = ∑1≤i′≤k

θi′ ⊗ θi′ − ∑k+1≤i′≤n

θi′ ⊗ θi′ .

(6.4) X connected =⇒ k is a constant on X .(6.5) X Riemannian manifold =⇒ index of X is n = dimX .

(7) Let X be a Riemannian manifold with dimX = n.(7.1) A basis (ei)1≤i≤n is orthonormal if

(3.1.10) (ei|ej) = δij.

(7.2) orthonormal frames in TxX is one-to-one to O(n):

(ei)1≤i≤n, (ei′)1≤i′≤n ←→ ei′ = aii′ ei with (ai

i′) ∈ O(n).

(7.3) Let

(3.1.11) O(X ) := (x, τx)|x ∈ X , τx orthonormal frame in TxX .

Then O(X ) is a principal fibre bundle with typical fibre O(n).

Let (X , g) be a 4-dimensional pseudo-Riemannian manifold of index 1.(1) X is called a Lorentz manifold or hyperbolic manifold and g is called

Lorentz.(2) Indices for coordinates:

Greek indices : α, β, γ, δ, · · · ,∈ 0, 1, 2, 3,Latin indices : i, j, k, ℓ, · · · ,∈ 1, 2, 3.

(3) A basis (eα)0≤α≤3 is called orthonormal if

(3.1.12) (eα|eβ) = 0 (∀ α = β), (e0|e0) = 1, (ei|ei) = −1 (∀ i).

(4) In terms of an orthonormal basis (θα)0≤α≤3,

(3.1.13) g = θ0 ⊗ θ0 − ∑1≤i≤3

θi ⊗ θi.

(5) The null cone or light-like cone:

(3.1.14) Cx := w ∈ TxX |gαβwαwβ = 0.

Cx consists of two half-cones: Cx = C+x ∪ C−x (See Figure 3.1).

3.1. PSEUDO-RIEMANNIAN STRUCTURES 133

Figure 3.1: Light-like cone

(5.1) If one of these half-cones is singled out and called the future half-cone C+x , then TxX is said to be time-oriented.

(5.2) If TxX can be time-oriented in a continuous fashion at each x ∈ X ,then X is said to be time-orientable.

(5.3) A hyperbolic manifold may be orientable but not time-orientable(See Figure 3.2):

V :=(x0, x1, x2) ∈ R3| − a ≤ x2 ≤ a

/(x0, x1, a) ∼ (−x0,−x1,−a)

and

X := V ×R, g = dx0 ⊗ dx0 − ∑1≤i≤3

dxi ⊗ dxi.

(6) w ∈ TxX is spacelike (outside the null cone) if gαβwαwβ < 0. w ∈ TxXis timelike (inside the null cone) if gαβwαwβ > 0.

(6.1) A timelike vector inside C+x is called future-directed.(6.2) A timelike vector inside C−x ia called post-directed.

(7) It is usual to choose a coordinate system (xα) = (x0, xi) on a hyperbolicmanifold in such a way that ∂/∂x0 is a timelike vector field the ∂/∂xi arespacelike vectors.


Figure 3.2: Orientable but not time-orientable

Theorem 3.1.1. Let u be a timelike vector field such that gαβuαuβ = 1. Then there exist(in a coordinate neighborhood U ⊂ X ) three spacelike vector fields which together with uform an orthonormal moving frame in U .

PROOF. WLOG, we may assume that u = a ∂∂x0 in a coordinate neighborhood

U . According to gαβuαuβ = 1, we have

1 = g00a2 =⇒ a2 =1

g00.

Write

U :=1a

.

Then

u =1U

∂

∂x0 , U2 = g00.

Setθ0 := uαdxα = gαβuβdxα = gα0u0dxα = Udx0 +

gi0U

dxi.

Defineγαβ := gαβ − uαuβ.

Then

gαβdxα ⊗ dxβ =(γαβ + uαuβ

)dxα ⊗ dxβ = θ0 ⊗ θ0 + γαβdxα ⊗ dxβ.

Howeverγαβdxα ⊗ dxβ =

(gαβ − uαuβ

)dxα ⊗ dxβ


=

(gij −

gi0gj0

U2

)dxi ⊗ dxj.

Since (X , g) is hyperbolic, it follows that gij < 0 and then

γαβdxα ⊗ dxβ = − ∑1≤i≤3

θi ⊗ θi

for some θ1, θ2, θ3 which forms an orthonormal moving frame in U .

(8) (eα)0≤α≤3 orthonormal frame =⇒

ηαβ := gα(eα, eβ) =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

= ηαβ.

(8.1) Define the complete Lorentz group by

(3.1.15) L(4) ≡ O(1, 3) =

L : (eβ) 7→ (aαβeα)

∣∣∣∣ L linear andaα

µηαβaβν = ηµν

.

(8.2) The orthochronous Lorentz group is

(3.1.16) L∗(4) :=(aα

β) ∈ L(4)∣∣∣∣a0

0 > 0

.

(8.3) An orthochronous Lorentz transformation preserves the sense of time-like vectors.

(8.4) The proper Lorentz group is

(3.1.17) L0(4) ≡ SO(1, 3) :=(aα

β) ∈ L(4)∣∣∣∣a0

0 > 0 and det(aαβ) = 1

.

(8.5) Spatial reflection:

s : (e0, ei) 7−→ (e0,−ei).

Time reflection:

t : (e0, ei) −→ (−e0, ei).

Observe that det(s) = det(t) = −1.(8.6) It can be shown that the complex Lorentz group is equal to

(3.1.18) L(4) = L0(4) ∪ sL0(4) ∪ tL0(4) ∪ stL0(4).

(8.7) L(4) is NOT connected.

Remark 3.1.2. ∃ bijection

(3.1.19) orthonormal frames in TxX ←→ L(4).

Define

(3.1.20) L(X ) := (x, τx)|x ∈ X , τx is an orthonormal frames in TxX .

Then, L(X ) is a principal fibre bundle over X with typical fibre and structuregroup L(4).


3.1.2. Geometry of submanifolds. f : X → Y differentiable mapping, (Y , g)pseudo-Riemannian manifold.

(1) Define

(3.1.21) (f∗g)x(v, w) := gf(x)(f′xv, f′xw

),

the first fundamental form of X or (X ,Y , f, g).

Theorem 3.1.3. (Y , g) Riemannian, f : X → Y immersion (of rank n = dimX ) =⇒f∗g defines a Riemannian structure on X .

PROOF. (i) f∗g is symmetric.(ii) f∗g is nondegenerate, since

0 = (f∗g)x(v, w) (∀ v) ⇐⇒ gf(x)(f′xv, f′xw) = − (∀ v)

⇐⇒ f′xw = 0 ⇐⇒ w = 0 (rank(f) = n).(iii) (f∗g)x(v, v) is positive definite: (f∗g)x(v, v) = gf(x) (f′xv, f′xv) ≥ 0.

Example 3.1.4. (1) f : R→ R2, x 7→ ( f 1(x), f 2(x)). Let gy(w, w) := (w1)2 +(w2)2.Then

(f∗g)x(v, v) = gf(x)(f′xv, f′xv

)= v2

[(d f 1

dx

)2

+

(d f 2

dx

)2].

The tensor f∗g is a metric on R only if f is od rank 1.(2) f : X → Y , dimX = 3, dimY = 4, and

f(x1, x2, x3) := (y0, y1, y2, y3) := (x1, x1, x2, x3).

Letgy(w, w) := (w0)2 − ∑

1≤i≤3(wi)2.

Then(f∗g)x(v, v) = −(v2)2 − (v3)2, (f∗g)x(u, v) = −u2v2 − u3v3.

The form (f∗g)x(u, v) is degenerate and therefore does NOT define a pseudo-Riemannian metric on X .

(3) Theorem 3.1.3 does NOT hold if Y is not Riemannian , or if f is not animmersion.

Let i :W → X be a submanifold of a hyperbolic manifold (so that dimX = 4).(1) W is called a null submanifold of X if i∗g is degenerate.

f : X → Y embedding, dimY = n, dimX = n− 1.(1) 0 = n ∈ T∗f(x)Y satisfying

⟨n, f′xv⟩ = 0, ∀ v ∈ TxX ,

defines a normal to f(X ), which is unique, up to a normalizing factor.


(2) If g is a Lorentz metric on Y , then

f∗g

Lotentzdegenerate

negative definite

if gαβnαnβ

< 0,= 0,> 0.

3.1.3. Existence of a pseudo-Riemannian structure. On a noncompact para-compact C1-manifold we give both Riemannian and Lorentzian structures. But inthe compact setting, the existence of a Lorentzian structure is related to its Euler-Poincare characteristic.

Theorem 3.1.5. A smooth (at least C1) manifold X can be given a Riemannian structureif and only if X is paracompact.

PROOF. A Riemannian manifold is a metric space and therefore is paracom-pact. Conversely, assume that X is paracompact. According to Theorem 2.5.1,there exists a partition of unity (θi)i∈I subordinate to an atlas ((Ui, φi))i∈I . Let gibe the 2-covariant tensor on Ui whose components with respect to the natural basisare

(gi)αβ = δαβ.

Then

∑i∈I

θigi

defines a Riemannian structure on X . This is because the sum of two positivedefinite quadratic forms is again a positive definite quadratic form.

The construction of Theorem 3.1.5 cannot be used to build a Lorentz structurebecause the sum of two quadratic forms of index 1 need to have index 1.

A line element or direction at x ∈ X is a 1-dimensional vector subspace ofTxX .

Theorem 3.1.6. On a paracompact C1-manifold, we have

(3.1.22) ∃ continuous linear element field ⇐⇒ ∃ Lorentz structure.

PROOF. =⇒: Since X is paracompact, by Theorem 3.1.5, ∃ Riemannian struc-ture g. Let u ∈ TxX be a unit vector in the line element at x (the vector u isuniquely determined up to a sign). Consider

gx(v, v) := 2[gx(v, u)]2 − gx(v, v), ∀ v ∈ TxX .

Locallygx(v, v) =

(2uαuβ − gαβ

)vαvβ.

Let e0 := u and choose e1, · · · , en−1 so that (eα)α forms an orthonormal frame at x.Hence u0 = u0 = 1, ui = ui = 0, gαβ = δαβ, and gx(v, v) = (v0)2 −∑1≤i≤n−1(vi)2.


⇐=: Let X be a Lorentz manifold with a Lorentz structure g. Let g defineRiemannian structure on X by Theorem 3.1.5. One of the eigenvector u of theeigenvalue problem

gαβuβ = λgαβuβ

is timelike. The timelike eigenvector determines a continuous line element fieldon X .

Theorem 3.1.7. (1) Any oncompact paracompact C1-manifold can be given a Lorentzstructure.

(2) A connected compact C1-manifold X can be given a Lorentz structure if χ(X ) =0.

(3) If a connected, compact, oriented manifold X can be given a Lorentz structure,then χ(X ) = 0.

3.1.4. Volume element and the star operator. Let X be an orineted pseudo-Riemannian manifold, dimX = n, and g a pseudo-Riemannian metric.

(1) Let

(3.1.23) det(g) := determinant of (gαβ) = ϵi1···in1···n gi11 · · · ginn.

(2) Under a transformation:

ek′ = Aik′ ei,

we havegk′ℓ′ = g(ek′ , eℓ′) = Ai

k′Ajℓ′gij.

Therefore

(3.1.24) det(g′) = |det A|2 det(g).

(3) If ek′ = Aik′ ei is an orientation-preserving transformation, then√|det(g′)| = |det A|

√|det(g)| =

√|det(g)|.

(4) Let

(3.1.25) dVg :=1n!

τi1···in dxi1 ∧ · · · ∧ dxin =√|det(g)|dx1 ∧ · · · ∧ dxn

with

(3.1.26) τi1···in := ϵ1···ni1···in τ1···n = ϵ1···n

i1···in

√|det(g)|.

(4.1) dVg is called the volume form or volume element of f on X .(4.2) dVg is independent of the orientation-preserving transformation.(4.3) If (θi)1≤i≤n is an orthonormal basis, then

(3.1.27) dVg = θ1 ∧ · · · ∧ θn.

(4.4) We have

τi1···in = g1i1 · · · gnin τ1···n = g1i1 · · · gnin√|det(g)| = sgn(det(g))√

|det(g)|ϵi1···in

1···n .


(5) The inner product on Λpx(X ):

(3.1.28)(dxi1 ∧ · · · ∧ dxip

∣∣dxj1 ∧ · · · ∧ dxjp)

:= det[(

dxik

∣∣∣∣dxjℓ)]

= ϵi1···ipk1···kp

gj1k1 · · · gjpkp .

By linearity, with

α =1p!

αi1···ip dxi1 ∧ · · · ∧ dxip , β =1p!

βi1···ip dxi1 ∧ · · · ∧ dxip ,

we have

(3.1.29) (α|β) = 1p!

αi1···ip βi1···ip = αI βI

with I = (1 ≤ i1 < · · · < ip ≤ n).(6) The star operator is

(3.1.30) ∗ : Λp(X ) −→ Λn−p(X ), β 7−→ ∗β,

defines the unique (n− p)-form ∗β satisfying

(3.1.31) α ∧ ∗β = (α|β)dVg, ∀ α ∈ Λp(X ).

(6.1) Choosing α = dx1 ∧ · · · ∧ dxp yields

dVg(dx1 ∧ · · · ∧ dxp|β) = dVg β1···p

=1

(n− p)!τ1···pip+1···in dx1 ∧ · · · ∧ dxp ∧ dxip+1 ∧ · · · ∧ dxin β1···p

= dx1 ∧ · · · ∧ dzp ∧[

1(n− p)!

(∗β)ip+1···in dxip+1 ∧ · · · ∧ dxin]

.

So

(3.1.32) (∗β)ip+1···in = τ1···pip+1···in β1···p =1p!

τi1···in βi1···ip .

(6.2) ∗ preserves the inner product up to sign:

(3.1.33) (∗α| ∗ β) = sgn(det(g))(α|β).

(6.3) The inverse ∗−1 is

(3.1.34) ∗−1 = (−1)p(n−p)sgn(det(g)) ∗ on Λp(X ).

Indeed, using (3.1.33),

dVg(β|α) = β ∧ ∗α = (−1)p(n−p) ∗ α ∧ β

and

dVg(β|α) = sgn(det(g))dVg(∗β| ∗ α) = sgn(det(g))dVg(∗α| ∗ β)

= sgn(det(g)) ∗ α ∧ ∗ ∗ β.

Thus ∗2 = (−1)p(n−p)sgn(det(g)) on Λp(X ).


(7) Define

δ := (−1)p ∗−1 d∗ : Λp(X ) −→ Λp−1(X ),

ω 7−→ (−1)(n−p)(p−1)−1sgn(det(g)) ∗ d ∗ ω.(3.1.35)

Then

(3.1.36) δ2 = 0.

(8) ∀ α, β ∈ Λp(X ) (one of supp(α) and supp(β) is compact), define theglobal inner product by

(3.1.37) [α|β] :=∫X(α|β)dVg =

∫X

α ∧ ∗β.

♣ Exercise: Proof (3.1.33) and (3.1.36).

Let T be a linear operator of degree s on the graded algebra Λ0(X ) of differ-ential forms with compact support. The metric transpose T′ of T is defined by

(3.1.38) [α|T′β] = [Tα|β], ∀ α ∈ Λp0 (X ), β ∈ Λ

p+s0 (X ).

Theorem 3.1.8. d and δ are the metric transposes of each other.

PROOF. For α ∈ Λp−10 (X ) and β ∈ Λ

p0 (X ), one has

[dα|β] =∫X

dα ∧ ∗β =∫X

d(α ∧ ∗β) + (−1)p∫X

α ∧ d ∗ β

= (−1)p∫X

α ∧ ∗(∗−1d ∗ β

)=

∫X

α ∧ ∗δβ = [α|δβ].

Thus d and δ are the metric transposes of each other. However, Theorem 3.1.8 does NOT hold for a compact manifold with bound-

ary.

♣ Exercise: Find the metric transpose of ∗.

3.1.5. Isometries. LetX andY be smooth manifolds with pseudo-Riemannianmetrics g and γ, respectively, and dimX = dimY = n.

(1) A mapping f : X → Y is called an isometry if f is diffeomorphic andf∗γ = g.

(2) Two manifolds are isometric if ∃ isometry of one onto the other.(3) f : X → Y is called a local isometry if ∀ x ∈ X ∃ neighborhoods U of x

and V of f(x) such that f : U → V is an isometry.(4) Let

(3.1.39) Iso(X ) := isometries of X onto X .

Then Iso(X ) is a group.


Figure 3.3: Theorem 3.1.9

(5) Given v ∈ TX . Then v generates a one-parameter pseudogroup of localisometries if and only if Lvg = 0.

Theorem 3.1.9. Assume f ∈ Iso(X ). Then ∀ chart (U ,φ) ∃ (U ′,φ′) such that φ(U ) =φ′(U ′) and

(3.1.40)[(φ−1)∗g

](xi) =

[(φ′−1)∗g

](xi), ∀ (xi) ∈ φ(U ) ⊂ Rn.

The converse is true locally.

PROOF. (i) Let (U ′,φ′) = (f(U ),φ f−1). Then φ(U ) = φ′(U ′) and

(φ′−1)∗g =((φ f−1)−1

)∗g = (φ−1)∗f∗g = (φ−1)∗g.

(ii) Conversely, f := φ′−1 φ : U → U ′ is a local isometry, since

f∗g = (φ′−1 φ)∗g = φ∗(φ′−1)∗g = φ∗(φ−1)∗g = g

by (3.1.40).

Any pseudo-Riemannian manifold that is isometric to the pseudo-Riemannianmanifold Rn with chart (Rn, 1) and metric

ds2 = ∑1≤i≤n

ϵidxi ⊗ dxi, ϵi := ±1,

is called a flat space.

(1) ϵi = +1 (for all i): Euclidean (En).


(2) Minkowski space is the flat space with metric

(3.1.41) ds2 = dx0 ⊗ dx0 − ∑1≤i≤n−1

dxi ⊗ dxi.

(3) A locally flat space is a space locally isometric to a flat space.

3.2. Linear connections

3.2.1. Linear connections. A linear connection on a smooth manifold X is amapping

(3.2.1) TX −→ T∗X ⊗ TX ∼= T1,1(X ), v 7−→ ∇v,

called the covariant derivative of v, satisfying

(3.2.2) ∇(v + w) = ∇v +∇w, ∇( f v) = d f ⊗ v + f∇v,

for all v, w ∈ TX and f ∈ C∞(X ).(1) The connection coefficients γj

ki are defined by

(3.2.3) ∇ei := γjkiθ

k ⊗ ej,

where (ej)1≤j≤n and (θk)1≤k≤n are dual base.(2) Then

∇v = ∇(viei) = dvi ⊗ ei + vi∇ei

=(

dvi + vjγikjθ

k)⊗ ei =

(ek(vi) + γi

kjvj)

θk ⊗ ei.

Define

(3.2.4) ∇v := ∇kviθk ⊗ ei ≡ vi;kθk ⊗ ei

so that

(3.2.5) vi;k = ek(vi) + γi

kjvj ≡ vi,k + γi

kjvj.

(3) The connection forms are

(3.2.6) ω ji := γj

kiθk

so that

(3.2.7) ∇v =(

dvi + ωijvj)⊗ ei.

(4) Under the change of basis

ei = aj′

i ej′ , θk = akℓ′θ

ℓ′ , aj′

i akj′ = δk

i , vi = aij′v

j′ .

Because

dvi = d(aij′v

j′) = aij′dvj′ + vℓ

′dai

ℓ′ = aiℓ′dvℓ

′+ vℓ

′eh′(ai

ℓ′)θh′ ,

we obtain

∇v =(

dvi + vjγikjθ

k)⊗ aj′

i ej′

=(

dvj′ + aj′

i vℓ′dai

ℓ′ + amℓ′v

ℓ′γikmaj′

i akh′θ

h′)⊗ ej′

=(

dvj′ + vℓ′γj′

h′ℓ′θh′)⊗ ej′

and then

(3.2.8) γj′h′ℓ′ = aj′

i akh′ a

mℓ′γ

ikm + aj′

iueh′(aiℓ′).

3.2. LINEAR CONNECTIONS 143

(5) In terms of the natural basis (∂/∂xi) we get

(3.2.9) ∇v =

(∂vi

∂xk + Γikℓvℓ

)dxk ⊗ ∂

∂xi ,

where Γikℓ are the connection coefficients in the natural basis (Christoffel

symbols). Under a change of natural basis

(3.2.10) Γj′h′ℓ′ =

∂xm

∂xℓ′∂xk

∂xh′∂xj′

∂xi Γikm +

∂xj′

∂xi∂

∂xh′

(∂xi

∂xℓ′

).

(6) On a Ck (k ≥ 2) manifold X a connection is said to be of class Cr if in allcharts of an atlas, Γk

ij are of class Cr. If r ≤ k− 2, the definition does notdepend on the atlas.

(7) v ∈ Ck−1 and ∇ ∈ Ck−2 =⇒∇v ∈ Ck−2 by (5).

Parallel translation. Let ∇ be a linear connection on X .

(1) The covariant derivative ∇uv v in the direction of u is

(3.2.11) ∇uv := (∇v)(u) or ∇uv(·) := ∇v(u, ·).

Thus

∇uv = uk(

ek(vi) + γikjvj)

ei =(

u(vi) + γikjukvj

)ei.

In particular,

(3.2.12) ∇ei ek = γjikej.

(2) ∇uv is linear in u over C∞(X ):

∇ f1u1+ f2u2 v = f1∇u1 v + f2∇u2 v.

(3) A vector v is said to be parallel along a curve C : t 7→ C(t), if

(3.2.13) ∇uv = 0, u := C′(

ddt

)≡ dC

dt, ui =

dCi

dt.

(3.1) The vector ∇uv is denoted Dv/dt.(3.2) The vector u = dC/dt is defined only at points along C. It can be

extended to a vector field on a neighborhood of any point of C (byusing a partition of unity). In terms of local coordinates, ∇uv is in-dependent of the extension.

♣ Exercise: Prove (3.2).

(4) An affine geodesic on X is a curve C : t 7→ C(t) such that

(3.2.14) ∇uu = λ(t)u, u =dCdt

.

(4.1) If ∇uu = 0, then the curve is a geodesic.


(4.2) In a local chart (U ,φ), with Ci(t) = φi C(t) and ui(t) = ui φ(t),

ui = u(φi) =ddt(φi C(t)) =

dCi(t)dt

, u(ui) =ddt(ui C(t)) =

dui(t)dt

.

Hence

(∇uu)i = u(ui) + ukΓkkjuj =

d2Ci(t)dt2 + Γi

kjdCk(t)

dtdCj(t)

dt.

(4.3) Under the change of parameter

s := τ(t), C(t) := C(s) = C τ(t), u =dCdt

, u =dCds

.

If ∇uu = λ(t()u, then

d2Ci

dt2 + Γikj

dCk

dtdCj

dt= λ(t)

dCi

dtand

ddt

(dCi

dsdτ

dt

)+ Γi

kjdCk

dsdCj

ds

(dτ

dt

)2= λ(t)

dτ

dtdCi

ds.

Hence(dτ

dt

)2 d2Ci

ds2 + Γikj

dCk

dsdCj

ds

(dτ

dt

)2=

[λ(t)

dτ

dt− d2τ

dt2

]dCi

ds.

We finally get

(3.2.15) ∇uu = λ(t)u, λ(t) :=[

λ(t)dτ

dt− d2τ

dt2

] (dτ

dt

)−2.

s is an affine parameter if λ(t) = 0, i.e., if τ′′(t)− λ(t)τ′(t) = 0.

♣ Exercise: If s is affine, so is as + b, where a, b ∈ R.

Theorem 3.2.1. x ∈ X , 0 = v ∈ TxX =⇒ ∃!, up to a change of parameter, maximalaffine geodesic C : t 7→ C(t), such that

C(0) = x, C(0) =dCdt

∣∣∣∣t=0

= v.

The covariant derivative is extended to tensors:(i) ∇v f = v( f ), ∀ f ∈ C1(X ),

(ii) ∇v(t + s) = ∇vt +∇vs,(iii) ∇v(t⊗ s) = ∇vt⊗ s + t⊗∇vs,(iv) ∇v commutes with the operation of contracted multiplication.

Recall the notion

Tp,q(X ) := ⊗pTX⊗⊗qT∗X , ∗Tp,q(X ) := ⊗pT∗X

⊗⊗qTX

so that ∗Tp,q(X ) ∼= Tq,p(X ).


(1) t ∈ ∗Tp,q(X ) =⇒∇t ∈ ∗Tp+1,q(X ) given by

(3.2.16) (∇t)(v, v1, · · · , vp, ω1, · · · , ωq) := (∇vt)(v1, · · · , vp, ω1, · · · , ωq).

(2) For α ∈ ∗T1,0(X ) = T∗X = T0,1(X ) and u ∈ ∗T0,1(X ) = TX = T1,0(X )=⇒∇v[α(u)] = ∇v [C(α⊗ u)] = C [∇v(α⊗ u)]

= C [∇vα⊗ u + α⊗∇vu] = (∇vα)(u) + α(∇vu).

Letting u = ei yields

(∇vα)i = v(αi)− α(

γjkivkej

)= v(αi)− γj

kivkαj.

Thus

(3.2.17) ∇vα = vk(

ek(αi)− γjkiαj

)θi

and

(3.2.18) ∇α =(

ek(αi)− γjkiαj

)θk ⊗ θi =

(dαi −ω j

iαj

)⊗ θi.

(3) In particular

(3.2.19) ∇θi = −ωij ⊗ θ j = −γi

kjθk ⊗ θ j, ∇vθi = −vkγi

kjθj.

(4) ∀ t ∈ ∗Tp,q(X ) we have

∇vt = ∇v

(ti1···iq j1···jp θ j1 ⊗ · · · ⊗ θ jp ⊗ ei1 ⊗ · · · ⊗ eiq

)= v(ti1···iq j1···jp)θ

j1 ⊗ · · · ⊗ θ jp ⊗ ei1 ⊗ · · · ⊗ eiq

+ ∑1≤k≤p

ti1···iq j1···jp θ j1 ⊗ · · · ⊗ ∇vθ jk ⊗ · · · ⊗ θ jp ⊗ ei1 ⊗ · · · ⊗ eiq

+ ∑1≤ℓ≤q

ti1···iq j1···jp θ j1 ⊗ · · · ⊗ θ jp ⊗ ei1 ⊗ · · · ⊗ ∇veiℓ ⊗ · · · ⊗ eiq .

Set∇αti1···iq j1···jp := ∇eα t

(ej1 , · · · , ejp , θi1 , · · · , θiq

).

Then

∇αti1···iq j1···jp = eα(ti1···iq j1···jp)− ∑1≤k≤p

γjαjk ti1···iq j1···jk−1 jjk+1···jp

+ ∑1≤ℓ≤q

γiℓαiti···iℓ−1iiℓ+1···iq j1···jp .(3.2.20)

(5) The formula for the components of the covariant derivative of a productis identical with the usual formula for the derivative of a product:

(3.2.21) ∇j(siktℓ) = tℓ∇jsi

k + sik∇jtℓ.

♣ Exercise: Verify (3.2.21).

Torsion and curvature.


(1) The torsion operation T∇ of ∇:

(3.2.22) T∇(u, v) := ∇uv−∇vu− [u, v].

(2) The curvature operation R∇ of ∇:

(3.2.23) R∇(u, v) := ∇u∇v −∇v∇u −∇[u,v].

(3) Note that

T∇(u, v) = −T∇(v, u), R∇(u, v) = −R∇(v, u), T∇( f u, gv) = f gT∇(u, v),

andR∇( f u, gv)hw = f ghR∇(u, v)w.

♣ Exercise: Verify (3).

(4) The torsion tensor of ∇:

(3.2.24) T(α, u, v) := α(T∇(u, v)) ∈ T1,2(X ).

(5) The curvature tensor of ∇:

(3.2.25) R(w, α, u, v) := α(R∇(u, v)w) ∈ T∗X ⊗ T1,2(X ).

(6) Define

Tikℓ := T(θi, ek, eℓ) = θi (∇ek eℓ −∇eℓ ek − [ek, eℓ]

)= γi

kℓ − γiℓk − ci

kℓ,(3.2.26)

Rijkℓ := R(ei, θ j, ek, eℓ) = θ j

(∇ek∇eℓ ei −∇eℓ∇ek ei −∇[ek ,eℓ ]ei

)= ek(γ

jℓi)− eℓ(γj

ki) + γjkmγm

ℓi − γjℓmγm

ki − cmkℓγ

jmi.(3.2.27)

Here

(3.2.28) [ek, el ] := cikℓei

with the structure coefficients cikℓ of the moving frame (ei)1≤i≤n.

(8) For f ∈ C∞(X ) and v ∈ TX we have

∇k∇ℓvj −∇ℓ∇kvj = Rijkℓvi − Ti

kℓ∇ivj,(3.2.29)

∇k∇ℓ f −∇ℓ∇k f = −Tikℓ∇i f .(3.2.30)

♣ Exercise: Verify (3.2.29) and (3.2.30).

(9) Define the torsion form and curvature form of ∇ by

Θi :=12

Tikℓθ

k ∧ θℓ,(3.2.31)

Ωji :=

12

Rijkℓθ

k ∧ θℓ.(3.2.32)

Theorem 3.2.2. (Cartan structural equation) We have

(3.2.33) Θi = dθi + ωiℓ ∧ θℓ, Ωj

i = dω ji + ω j

m ∧ωmi.


PROOF. From the identity

dθ(u, v) = Lu(θ(v))−Lv(θ(u))− θ([u, v])

we obtaindθi(ek, eℓ) = −θi([ek, eℓ]) = −ci

kℓand hence

dθi = −12

cikℓθ

k ∧ θℓ.

Alsoωi

ℓ ∧ θℓ = γikℓθ

k ∧ θℓ =12(γi

kℓ − γiℓk)θ

k ∧ θℓ.

Thereforedθi + ωi

ℓ ∧ θℓ =12(γi

kℓ − γiℓk − ci

kℓ)θk ∧ θℓ = Θi.

For the second equation in (3.2.33),

ω jm ∧ωm

i = γjkmγm

ℓiθk ∧ θℓ =

12(γj

kmγmℓi − γj

ℓmγmki)θ

k ∧ θℓ,

and then

dω ji(ek, eℓ) =

(d(γj

miθm))(ek, eℓ) =

(dγj

mi ∧ θm + γjmidθm

)(ek, eℓ)

= ek(γjmi)δ

mℓ − eℓ(γj

mi)δmk − γj

micmkℓ = ek(γ

jℓi)− eℓ(γj

ki)− cmkℓγ

jmi.

Thus Ωji = dω j

i + ω jm ∧ωm

i. (10) Differentiation of the Cartan structural equations gives

dΘk = dωki ∧ θi −ωk

i ∧ dθi

= (Ωki −ωk

m ∧ωmi) ∧ θi −ωk

i ∧ (Θi −ωim ∧ θm)(3.2.34)

= Ωki ∧ θi −ωk

i ∧Θi,

and

dΩki = dωk

ℓ ∧ωℓi −ωk

ℓ ∧ dωℓi

= (Ωkℓ −ωk

m ∧ωmℓ) ∧ωℓ

i −ωkℓ ∧ (Ωℓ

i −ωℓm ∧ωm

i)(3.2.35)

= Ωkℓ ∧ωℓ

i −ωkℓ ∧Ωℓ

i.

Then

13!

∑(jℓi)

∂jTkℓi

dxj ∧ dxℓ ∧ dxi =13!

∑(jℓi)

(Rik

jℓ − ΓkjmTm

ℓi)

dxj ∧ dxℓ ∧ dxi

where ∂j := ∂/∂xj and ∑(jℓi) denotes the sum over cyclic permutations of(jℓi). On the other hand

∑(jℓi)∇jTk

ℓi = ∑(jℓi)

(∂jTkℓi + Γk

jmTmℓi − Γm

jℓTkmi − Γm

jiTkℓm)

= ∑(jℓi)

(∂jTkℓi + Γk

jmTmℓi + Tm

jiTkmℓ)

so

(3.2.36) ∑(jℓi)

Rik

jℓ = ∑(jℓi)

(∇jTkℓi − Tm

jiTkmℓ).


Similarly

(3.2.37) ∑(jkℓ)∇jRi

mkℓ = ∑

(jkℓ)Th

kjRim

hℓ.

♣ Exercise: Complete the proof of (3.2.36) and (3.2.37).

3.2.2. Levi-Civita connections/Riemannian connections. The most importantconnection is the Levi-Civita connection.

Theorem 3.2.3. On a pseudo-Riemannian manifold (X , g) there exists a unique linearconnection ∇ = ∇g, called the Levi-Civita connection or Riemannian connection,such that

T∇ = 0, ∇g = 0.

PROOF. According to (3.2.33),

T∇ = 0 ⇐⇒ dθi + ωiℓ ∧ θℓ = 0 ⇐⇒ γi

ℓj − γijℓ = ci

ℓj.

Similarly

∇g = 0 ⇐⇒ dgij = ωℓigℓj + ωℓ

jgiℓ ⇐⇒ ek(gij) = γℓkigℓj + γℓ

kjgiℓ.

Then

γmik =

12

gjm[ek(gij) + ei(gjk)− ej(gki)]−12

(cm

ki + gjmcℓkj + gjmgℓkcℓij)

.

Defining γmik := gmjγjik and cmik := gmjc

jik yields

γmik = −12(cmki + cikm + ckim) +

12[ek(gim) + ei(gmk)− em(gki)].

Thus, the connection is determined by the metric g.

In the case of a natural frame,

(3.2.38) Γmik =

12

gjm(∂kgij + ∂igjk − ∂jgki), Γmik = Γm

ki .

In the case of an orthonormal frame

(3.2.39) γmik = −12(cmki + cikm + ckim), γmik = −γkim.

Let∇ be the Levi-Civita connection of a pseudo-Riemannian manifold (X , g).(1) ∇dVg = 0.(2) The parallel translation defined by the Levi-Civita connection preserves

the scalar product. Indeed, let u, v be parallel along a curve C : t 7→ C(t).Then

∇Cu = ∇Cv = 0 along C.Moreover

ddt⟨u, v⟩ = ⟨∇Cu, v⟩+ ⟨u,∇Cv⟩ = 0.


Curvature tensor, Ricci tensor, and scalar curvature. Let (X , g) be a pseudo-Riemannian manifold with the Levi-Civita connection ∇.

(1) The components of the curvature tensor of∇, called the Riemann tensor,satisfy

(1.1) Rijkℓ = −Ri

jℓk.

(1.2) ∑(ikℓ) Rijkℓ = 0.

(1.3) (Bianchi identity) ∑(mkℓ)∇mRijkℓ = 0.

(1.4) Rijkℓ = −Rjikℓ, where Rijkℓ := gjmRim

kℓ.(1.5) Rijkℓ = Rkℓij.

(2) The Ricci tensor is a contraction of the curvature tensor:

(3.2.40) Rik := Rijkj.

Clearly that

(3.2.41) Rij = Rji, Rik = ∂kΓjji − ∂jΓ

jki + Γj

kmΓmji − Γj

ℓmΓmki .

(3) The scalar curvature of g is

(3.2.42) R := gijRij.

From

0 = ∇mRijkℓ +∇kRi

jℓm +∇ℓRi

jmk

we get

0 = ∇mRik −∇kRim +∇jRijmk, 0 = ∇iRik −∇kR +∇jRj

m = 2∇iRik −∇kR.

Thus (contracted Bianchi identity)

(3.2.43) ∇iRik =12∇kR,

or

(3.2.44) ∇i(

Rik −R2

gik

)= 0.

We call Rij − R2 gij the Einstein tensor of g.

♣ Exercise: Show that

Γjjk =

∂

∂xk ln |√|det(g)|.

Theorem 3.2.4. A pseudo-Riemannian manijfold (X , g) is locally flat⇐⇒ Rm = 0.


♣ Exercise: Consider Hamilton’s cigar soliton or Witten’s black hole (Σ, g):

g :=dx⊗ dx + dy⊗ dy

1 + x2 + y2 .

In the polar coordinates (r, θ), we have

g =dr⊗ dt + r2dθ ⊗ dθ

1 + r2 , r =√

x2 + y2.

(1) Show that the scalar curvature is R = 4/(1 + r2).(2) Show that the area element is dVg = dx ∧ dy/(1 + r2).(3) Show that the geodesics C(t) = (r(t), θ(t)) satisfy

θ′′ +2

r(1 + r2)θ′r′ = 0 = r′′ − r

1 + r2

[(r′) + θ′(r)

].

Then we have

θ(t) = a + bt + b∫ t

0

dτ

r2(τ).

♣ Exercise: Consider the Riemannian metric

g = dx⊗ dx + f (x)2dy⊗ dy

on R2. Find the scalar curvature of g.

3.2.3. Second fundamental form. Let X be a smooth n-dimensional mani-fold, and Z ⊂ X a subset.

(1) Z is a submanifold of X if for any x ∈ Z ⊂ X there exists a chart (U , φ)of X about x, such that

φ : U ∩ Z −→ Rq × a, x 7−→ (x1, · · · , xq, aq+1, · · · , an),

where a ∈ Rn−q is a fixed point. Then (U , φ), where U := U ∩ Z andφ : U → Rq given by φ(x) := (x1, · · · , xq), form an atlas on Z .

(2) If Z has already a manifold structure, it is called a submanifold of X ifit can be given a submanifold structure that is equivalent to the existingstructure.

Theorem 3.2.5. Assume

Z := x ∈ X | f α(x) = 0, 1 ≤ α ≤ p ⊂ X ,

where f α(x) are differentiable functions of x and the mapping

X −→ Rp, x 7−→ ( f 1(x), · · · , f p(x))

is of rank p for all x ∈ Z . Then Z is a submanifold of X of dimension n− p.

PROOF. Given x0 ∈ Z and (U , φ) a chart of X about x0. Since he mappingx 7→ ( f 1(x), · · · , f p(x)) is of rank p at x0, it follows that there exists a p × pmatrix in [∂ f α/∂xi]1≤α≤p,1≤i≤n such that its determinant at x0 is nonzero, say


[∂ f α/∂xi]1≤α,i≤p. By the implicit function theorem, there exists an open subsetU ′ ⊂ U of x0 such that

xi = gi(xp+1, · · · , xn), 1 ≤ i ≤ p.

Here gi are differentiable in U ′. Define

x′ i := gi(xp+1, · · · , xn) 1 ≤ i ≤ p, x′p+1 := xp+1, · · · , x′n := xn.

Then the local chart (U ′, φ′) is such that

φ′(U ′ ∩ Z) = (0, · · · , 0, xp+1, · · · , xn).

Hence dimZ = n− p.

Example 3.2.6. (1) Sn−1 is a submanifold of Rn:

Sn−1 =

x ∈ Rn| ∑

1≤i≤n(xi)2 − 1 = 0

−→ R, x = (x1, · · · , xn) 7−→ ∑

1≤i≤n(xi)2− 1.

Since the above map has rank 1, it follows that dim Sn−1 = n− 1.(2) The cone

Cn−1 :=

x ∈ Rn|(x1)2 − ∑

2≤i≤n(xi)2 = 0

is NOT a submanifold of Rn:

f : Cn−1 −→ R, x = (x1, · · · , xn) 7−→ (x1)2 − ∑2≤i≤n

(xi)2.

Since

rankx( f ) =

0, x = 0,1, x = 0,

it follows that dim(Cn−1 \ 0) = n− 1.

We say a pair (Y , f) is an immersion if the differentiable mapping f : Y → Xis of rank q = dimY for every y ∈ Y .

(1) An immersion id NOT necessarily injective, so that f(Y) is NOT neces-sarily a manifold.

(2) f′y : TyY → Tf(y)X is injective.(3) An injective immersion is an embedding.

(3.1) The set f(Y) with the differentiable structure induced by the embed-ding f is a manifold:

(F(Vi, ψi F−1),

where (Vi, ψi) is an atlas of Y , and F : Y → f(Y) is a diffeomor-phism.

(3.2) The manifold structure induced by f on f(Y) may NOT be equiva-lent to a submanifold structure on f(Y).

(3.3) If f(Y) has a submanifold structure equivalent to the manifold struc-ture induced by f, then the embedding is said to be regular.


♣ Caution: In other books, our “embedding” is called “immersion” while“regular embedding” is called “embedding”.

We say a pair (Y , f) is an submersion if the differentiable mapping f : Y → Xis of rank n = dimX for every y ∈ Y (Then dimY ≥ n).

Let S ⊂ (X , g) be a submanifold of dimension q of a pseudo-Riemannianmanifold (X , g), and ∇ its Levi-Civita connection.

(1) The induced metric g on S is g := ι∗g, ι : S → X .(2) For any x ∈ S , TxS ⊂ TxX is a p-dimensional vector subspace.(3) Suppose that TxS is NOT null for g. Let T⊥x S be the g-orthogonal com-

plement of TxS in TxX .(4) If u ∈ TxS and v ∈ TxS , we can define ∇uv. Write

∇∥uv := component along TxS = tangent component of ∇uv,

∇⊥u v := component along T⊥x S = normal component of ∇uv.

Theorem 3.2.7. (a) ∇∥uv is the covariant derivative of v in the Levi-Civita connection ofthe metric g on S induced by g on X :

(3.2.45) ∇∥uv = ∇uv.

(b) ∇⊥u v = ∇⊥v u and ∇⊥u v depends only on the vectors ux and vx.

PROOF. Let u, v, w ∈ TS and f , h ∈ C∞(S).(a) Since

∇u( f v) = (u f )v + f∇uvit follows that

∇∥u( f v) = (u f )v + f∇∥uv, ∇⊥u ( f v) = f∇⊥u v.

Since∇uv−∇vu = [u, v],

we have∇∥uv−∇∥vu = [u, v], ∇⊥u v−∇⊥v u = 0.

Moreover

g(∇∥uv, w) + g(v,∇∥uw) = g(∇uv, w) + g(v,∇uw) = u(g(v, w)) = u(g(v, w)).

By the uniqueness of the Levi-Civita connection, we get ∇∥ = ∇.(b) We have proved in (1) that ∇⊥u v = ∇⊥v u. Moreover

∇ f uv = f∇uv = f(∇uv +∇⊥u v

)= ∇ f uv + f∇⊥u v.

Hence∇⊥f uv = f∇⊥u v, ∇⊥f uhv = f h(∇⊥u v).

Writing u = ui∂i and v = vi∂i yields ∇⊥u v = uivj∇⊥∂i∂j.


(5) The symmetric mapping

(3.2.46) kx : TxS × TxS −→ T⊥x S , (ux, vx) 7−→ (∇⊥u v)x = ∇⊥ux vx

is called the second fundamental form of S in X .(5.1) dimX = n = p+ 1 =⇒S is a hypersurface ofX , T⊥x S is 1-dimensional.

Hence T⊥x S is spanned by nx, where n is a unit normal to S . Write

(3.2.47) kx(ux, vx) = Kx(ux, vx)nx

and call K the extrinsic curvature of S in X . Thus K ∈ T0,2S . Wesay trg(K) the mean extrinsic curvature of S in X . When g is a Rie-mannian metric (resp. Lorentzian metric), then (S , trg(K) = 0) iscalled a minimal hypersurface (resp. maximal hypersurface).

(5.2) Let S ⊂ X be an embedding hypersurface and dimX = n = p + 1.Write f : S → X , s 7→ x := f (s). The unit normal n ∈ T∗f(s)Xis defined (up to sign, that is, + if g is Riemannian and − if g isLorentzian) by

⟨n, (Tsf)v⟩ = 0 for any v ∈ TsS , gαβnαnβ = ±1.

Recall that Lng denotes the Lie derivative of g with respect to anyvector field equal to n at x = f(s). Set (Lng)∥ the projection of Lngon Tf(s)f(S). Locally we have

(3.2.48) (Lng)∥αβ = hγα hδ

β(Lng)γδ, hαβ := gαβ ∓ nαnβ, hβα := hαγgγβ.

We claim that

(3.2.49) K = ∓12(Lng)∥.

Indeed, we let (xα)0≤α≤p and (si)1≤i≤p be coordinates of X and Srespectively. Then

(xα) = (x0, xi) = ( f α(s)) = (0, si)

so that x0 = 0 in the local equation of f(S). For v ∈ TsS write

v = vu ∂

∂si = vi ∂

∂xi .

Hence (0, vi) = (Ts f )v ∈ Tf(s)f(S), n = (n0, 0) with n20 = ±(g00)−1.

By definition, where (ei)1≤i≤p is the natural basis in Tf(s)f(S),

K(ei, ej)nα = (∇⊥eiej)

α = ±nαnβ(∇ei ej)β, ∀ ei, ej ∈ Tf(s)f(S),

because, for any w ∈ Tf(s)X ,

(w∥)α + (w⊥)α = wα = δαβwβ = (hα

β ± nαnβ)wβ.

Therefore

Kij = K(ei, ej) = ±nβ(∇ei ej)β = ±nβΓβ

ij = ±n0Γ0ij.

On the other hand,

−12(Lng)∥ij = −

12

hαi hβ

j (Lng)αβ = −12(Lng)ij = −

12(ni;j + nj;i) = Γ0

ijn0.


(5.3) (Gauss-Codazzi relations) Let (S , g, K) → (X , g) be a hypersurfacein a Riemannian manifold, and set Gµν := Rµν − 1

2 Rgµν. Then

(3.2.50) 2G00 = −R + Ki

jKji − (trgK)2, Gi

0 = ∇jKij −∇itrgK.

In fact, let (xα)0≤α≤p be local coordinates in X with x0 = 0 the localequation of S . The Gauss normal coordinate metric is

ds2 = dx0 ⊗ dx0 + gij(xα)dxi ⊗ dxj.

Then, where 1 ≤ i, j, k, ℓ ≤ p,

g00 = 1, g0i = 0, gij|S = gij, g00 = 1, g0i = 0, gijgjk = δik

and

Γ000 = Γ0

0i = Γi00 = 0, Γℓ

0k = −gℓmΓ0mk, Γi

jk|S = Γijk.

In order to compute

G00 =

12(R00 − gijRij), Gi

0 = g00R0i

we need to find

R00 = ∂0Γkk0 + Γk

0mΓmk0, R0j = ∂jΓk

k0 − ∂kΓkj0 + Γk

jmΓmk0 − Γk

kmΓmj0

and

Rij = Rij − ∂0Γ0ij + Γ0

jmΓm0i + Γk

j0Γ0ki − Γk

k0Γ0ji, Γ0

ij = Kij = −12

∂0gij.

From

∂0Γℓ0k = −∂0(gℓmΓ0

mk) = −2Kℓj K j

k − gℓm∂0Kmk

and

∂jKkk0 = −∂j(gkmΓ0

mk) = −∂jgkmΓ0mk − gkm∂jΓ0

mk = Kmk∂jgkm − gkm∂jKmk,

we obtain

2G00 = R00 − gijRij (gijΓk

j0Γ0ki = gijΓ0

ikΓk0j)

= R00 −(

R− gij∂0Γ0ij + gijΓ0

jmΓm0i + gijΓk

j0Γ0ki − gijΓk

k0Γ0ji

)= ∂0Γk

k0 + Γk0mΓm

k0 − R + gij∂0Γ0ij − 2gijΓ0

jmΓm0i + gijΓ0

ijΓkk0

= −2KjkKk

j − gkm∂0Kmk + gij∂0Kij − R + Γk0mΓm

k0 − 2gijΓ0jmΓm

0i + gijΓ0ijΓ

kk0

= −2KjiKi

j − R + KkmKm

k + 2gijKjmKmi − gijKijKk

k

= −R + KijKj

i − (Kii)2

and

Gi0 = g00R0i = ∂iΓk

k0 − ∂kΓki0 + Γk

imΓmk0 − Γk

kmΓmi0

= −∂iK jj + ∂kKk

i − ΓkimKm

k + ΓkkmKm

i

= ∂jK ji + Γj

jkKki − Γk

jiKjk − ∂iK j

j = ∇jK ji −∇itrgK.


♣ Exercise: Compete the proof of (3.2.50). When (X , g) is Lorentzian, we takethe g in the form

ds2 = −dx0 ⊗ dx0 + gij(xα)dxi ⊗ dxj.

In this case gαβnαnβ = −1 and Kij = Γ0ij =

12 ∂0gij. Please find the similar formulas

for G00 and Gi

0. Actually, up to sign, it is the same as (3.2.50).

When (X , g) is Einstein, that is, Gµν = 0, we see that (3.2.50) becomes

(3.2.51) R− |K|2g + (trgK)2 = 0, ∇jKji −∇itrgK = 0.

In this case, we call (S , g, K) is an initial data set of the Einstein manifold (X , g).The Cauchy problem of Einstein’s vacuum equation is, given an initial data setwith (3.2.51), to find a space-time (X , g) which satisfies Gµν = 0. Note that (3.2.51)is a necessary condition for the Cauchy problem.

♣ Exercise: Show that the Schwarzschild metric

g = −(

1− 2Mr

)dt⊗ dt +

(1− 2M

r

)−1dr⊗ dr + r2(dθ ⊗ dθ + sin2 θ dφ⊗ dφ)

defined on R2 × S2 ∋ (t, r, θ, φ), is a solution of the Einstein vacuum equationRµν = 0.

3.2.4. Differential operators. Let (X , g) be a pseudo-Riemannian manifoldof dimension n. A mapping D : C∞(X ) → C∞(X ) is called a linear differentialoperator of order m if on each local chart (U ,φ) there exist aj ∈ C∞(U ) withj = (j1, · · · , jn) ∈ Zn

≥0 such that

(3.2.52) (D f )(x) := ∑0≤|j|≤m

ajDj( f φ−1)(φ(x)), f ∈ C∞(X ), x ∈ U ,

where

Dj :=(

∂

∂x1

)j1· · ·(

∂

∂xn

)jn

A linear differential operator is local:

f1|V = f2|V =⇒ (D f1)|V = (D f2)|V .

Therefore it defines an operator on germs of functions. Moreover,

supp(D f ) ⊂ supp( f ).

A linear differential operator D is invariant with respect to a diffeomorphism Φof X if

(3.2.53) Φ∗(D f ) = D(Φ∗ f ), f ∈ C∞(X ).

There are some natural linear differential operators.


Exterior derivative d. For ω = 1p! ωi1···ip dxi1 ∧ · · · ∧ dxip we have

dω =1p!

∂jωi1···ip dxj ∧ dxi1 ∧ · · · ∧ dxip

=1p!∇jωi1···ip dxj ∧ dxi1 ∧ · · · ∧ dxip(3.2.54)

=1

p!(p + 1)!ϵ

ji1···ipk1···kp+1

∇jωi1···ip dxk1 ∧ · · · ∧ dxkp+1

so that

(3.2.55) dω =1

(p + 1)!(dω)ji1···ip dxj ∧ dxi1 ∧ · · · ∧ dxip

with

(3.2.56) dωji1···ip ≡ (dω)ji1···ip = p∇jωi1···ip .

The operator δ. For ω = 1p! ωi1···ip dxi1 ∧ · · · ∧ dxip we have (recall (3.1.25))

d(∗ω) =1

p!(n− p)!τi1···in∇jω

i1···ip dxj ∧ dxip+1 ∧ · · · ∧ dxin

=1

(n− p + 1)!p!(n− p)!ϵ

jip+1···inkp ···kn

τi1···ipip+1···in∇jωi1···ip dxkp ∧ · · · ∧ dxkn .

Then

δω =(−1)p(−1)(n−p+1)(p−1)sgn(det(g))

(p− 1)!(n− p + 1)!(n− p)!p!τkp ···knk1···kp−1

·ϵkp ···knjip+1···in τi1···in∇jωi1···ip dxk1 ∧ · · · ∧ dxkp−1

=(−1)p(−1)(n−p+1)(p−1)

(p− 1)!p!(n− p)!ϵ1···n

jip+1···ink1···kp−1ϵ

i1···in1···n ∇

jωi1···ip dxk1 ∧ · · · ∧ dxkp−1

=(−1)p(−1)(n−p+1)(p−1)(−1)(p−1)(n−p)

(p− 1)!p!(n− p)!ϵi1···in

jk1···kp−1ip+1···in

·∇jωi1···ip dxk1 ∧ · · · ∧ dxkp−1

=−1

(p− 1)!p!ϵ

i1···ipjk1···kp−1

∇jωi1···ip dxk1 ∧ · · · ∧ dxkp−1

=−1

(p− 1)!∇jωjk1···kp−1 dxk1 ∧ · · · ∧ dxkp−1

so that

(3.2.57) δωk1···kp−1 = (δω)k1···kp−1 = −∇jωjk1···kp−1 .

The divergence div. For any v ∈ TX we have

div(v) = ∇ivi = ∂ivi + Γiikvk = ∂ivi + vk∂k ln

√|det(g)|

=1√|det(g)|

∂i

(√|det(g)|vi

).(3.2.58)


(1) For any v ∈ TX we have

(3.2.59) d ∗ v = div(v)dVg.

Indeed,

∗v =1

(n− 1)!τi1···in vi1 dxi2 ∧ · · · ∧ dxin

=1

(n− 1)!

√|det(g)|ϵ1···n

i1···in vi1 dxi2 ∧ · · · ∧ dxin

=√|det(g)| ∑

1≤i≤n(−1)i−1vidx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn

and

d ∗ v = ∑1≤i≤n

(−1)i−1∂k

(√|det(g)|vi

)dxk ∧ dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn

= ∂k

(√|det(g)|vk

)dx1 ∧ · · · ∧ dxn

=√|det(g)|div(v)dx1 ∧ · · · ∧ dxn = div(v)dVg.

(2) Then

(3.2.60)∫

∂C∗v =

∫C

d ∗ v =∫C

div(v)dVg.

(3) For any v ∈ TX we have

(3.2.61) ∗ v = ιvdVg.

The Laplacian ∆. It is defined by

(3.2.62) ∆ := dδ + δd : Λp(X ) −→ Λp(X ).

(1) ω is harmonic if ∆ω = 0.(2) ∆ is self-adjoint (using the notation (3.1.37)):

[∆α|β] = [δα|δβ] + [dα|dβ] = [α|∆β], ∀ α, β ∈ Λpc (X ).

(3) If (X , g) is a Riemannian manifold, then ∆ ≥ 0:

[∆ω|ω] = [δω|δω] + [dω|dω] ≥ 0, ∀ ω ∈ Λpc (X ).

(4) We have

(3.2.63) ∗ ∆ = ∆∗, δ∆ = ∆δ, d∆ = ∆d.

(5) For ω = 1p! ωi1···ip dxi1 ∧ · · · ∧ dxip , we have

∆ωi1···ip = (∆ω)i1···ip = −gij∇i∇jωi1···ip

+ ∑1≤q≤p

(−1)qRiqhωhi1···iq ···ip

+ 2 ∑1≤r<q≤p

(−1)r+qRhiq

jir ωjhi1···ir ···iq ···ip

.

For f ∈ C∞(X ) one has

(3.2.64) ∆ f = −gij∇i∇j f = − 1√|det(g)|

∂i

(gij√|det(g)|∂j f

).


Theorem 3.2.8. If Φ ∈ Diff(X ), then

Φ ∈ Iso(X ) ⇐⇒ Φ∗(∆ f ) = ∆(Φ∗ f ), ∀ f ∈ C∞(X ).

PROOF. (1) Assume that Φ is an isometry. Let (U ,φ) be a local chart aboutx and choose (U ′,φ′) := (Φ(U ),φ Φ−1) as a local chart about x′ := Φ(x). ByTheorem 3.1.9 we get

gi j = g′i j ⇐⇒ gij φ−1 = gi′ j′ φ′−1 on φ(U ) = φ′(U ′).

Here g := (φ−1)∗g, g′ := (φ′−1)∗g are the induced metrics on φ(U ) = φ′(U ′) ⊂Rn, while

unprime components (U ,φ)

prime components (U ′,φ′)circumflexed components Rn.

Therefore

gij(x) = (gi′ j′ φ′−1 φ)(x) = (gi′ j′ Φ)(x) = gi′ j′(x′)

Γkij(x) = Γk′

i′ j′(x′),∂ f

∂xj′ (x′) =∂( f Φ)

∂xj (x),∂2 f

∂xi′∂xj′ (x′) =∂2( f Φ)

∂xi∂xj (x).

From

∆(Φ∗ f )(x) = ∆( f Φ)(x) = −gij(x)∂2( f Φ)

∂xi∂xj (x) + gij(x)Γℓij(x)

∂( f Φ)

∂xℓ(x)

and

Φ∗(∆ f )(x) = (∆ f )(x′) = −gi′ j′(x′)∂2 f

∂xi′∂xj′ (x′) + gi′ j′(x′)Γℓ′i′ j′(x′)

∂ f∂xℓ′

(x′),

we see that Φ∗(∆ f ) = ∆(Φ∗ f ).(2) Assume Φ∗(∆ f ) = ∆(Φ∗ f ). Then

0 = ∆(Φ∗ f )(x)−Φ∗(∆ f )(x) = −[

gij(x)− gi′ j′(x′)] ∂2( f Φ)

∂xi∂xj (x)

+[

gij(x)Γℓij(x)− gi′ j′(x′)Γℓ′

i′ j′(x′)] ∂( f Φ)

∂xℓ(x)

so that gij(x) = gi′ j′(x′) or g = Φ∗g.

3.2.5. Conformal transformation. Let (X , g) be a Riemannian manifold ofdimension n and ∇ be its Levi-Civita connection. Recall from (3.2.38), (3.2.27),(3.2.41), and (3.2.42) that

Γkij =

12

gkℓ(∂igℓj + ∂jgiℓ − ∂ℓgij),

Rijkℓ = ∂kΓj

ℓi − ∂ℓΓjki + Γj

kmΓmℓi − Γj

ℓmΓmki ,

Rik = Rijkj = ∂kΓj

ji − ∂jΓjki + Γj

kmΓmji − Γj

ℓmΓmki ,

R = gijRij.

3.3. VARIATIONS IN RIEMANNIAN GEOMETRY 159

We say two Riemannian metrics g and g are conformal, written as g ∼ g, ifg = f g for some f ∈ C∞(X ). Then f must be positive everywhere. It is clear that∼ is an equivalent relation. The equivalence class of ∼ is denoted

(3.2.65) [g] := g|g ∼ g = f g| f ∈ C∞(X ).

For g ∈ [g], we usually write g = e2ug for some u ∈ C∞(X ). The correspond-ing Levi-Civita connection is ∇. Then

Γkij = Γk

ij + δkj ∂iu + δk

i ∂ju− gkℓgij∂ℓu.

Proposition 3.2.9. If g = e2ug then

(3.2.66) R = e−2u[

R + 2(n− 1)∆u + (n− 1)(n− 2)|∇u|2]

.

Moreover

Rij = Rij + (n− 2)(∇i∇ju−∇iu∇ju +

12|∇u|2gij

)+

(∆u +

n− 22|∇u|2

)gij.(3.2.67)

♣ Exercise: Prove (3.2.66) and (3.2.67).

When n = 2, we have

R = e−2u (R + 2∆u) , Rij = Rij + ∆u gij.

The Yamabe problem, proposed by Yamabe in 1960, is to find a conformalmetric g ∈ [g] such that R is constant. Letting

u =2

n− 2ln φ, φ ∈ C∞(X ) and φ > 0

in (3.2.66) yields

(3.2.68) ∆φ +n− 2

4(n− 1)Rφ =

n− 24(n− 1)

Rφn+2n−2 .

Theorem 3.2.10. (Yamabe-Trudinger-Aubin-Schoen) If (X , g) is a closed Riemann-ian manifold of dimension n ≥ 3, then there is a conformal metric g ∈ [g] such that R isa constant.

3.3. Variations in Riemannian geometry

In Riemannian geometry, geodesics and minimal submanifolds can be used tostudy the geometry and topology of manifolds. Let (M, g) be an n-dimensionalRiemannian manifold and ∇ its Lrvi-Civita connection.


3.3.1. First variation of arc length. Given a path γ : [a, b] →M, its length isdefined by

(3.3.1) Lg(γ) :=∫ b

a|γ(u)|gdu.

The distance function is defined by

(3.3.2) dg,p(x) := dg(p, x) := infγ

Lg(γ),

where the infimum is taken over all paths γ : [0, 1] → M with γ(0) = p andγ(1) = x. A geodesic segment is minimal if its length is equal to the distancebetween the two endpoints.

Let γr : [a, b] → M, r ∈ r ⊂ R, be a 1-parameter family of paths. We definethe map Υ : [a, b]× r→M by

(3.3.3) Υ(s, r) := γr(s).

We define the vector fields R and S along Υ by

(3.3.4) R := Υ∗

(∂

∂r

), S := Υ∗

(∂

∂s

).

We call R the variation vector field and S the tangent vector field. More precisely,the map Υ induces a map between tangent spaces at each point (s, r) ∈ [a, b]× r:

Υ∗,(s,r) : T(s,r) ([a, b]× r) −→ Tγr(s)M.

Then

S(γr(s)) = Sγr(s) = Υ∗,(s,r)

((∂

∂s

∣∣∣s

), 0)

= Υ∗,(s,r)

(∂

∂s

∣∣∣s

),

R(γr(s)) = Rγr(s) = Υ∗,(s,r)

(0,(

∂

∂r

∣∣∣r

))= Υ∗,(s,r)

(∂

∂r

∣∣∣r

).

Thus,

S, R ∈ C∞(

Υ ([a, b]× r) , TM∣∣∣Υ([a,b]×r)

)By the above notation, we have

∂

∂r

∣∣∣r⟨S(γr(s)), S(γr(s))⟩g =

∂

∂r

∣∣∣r⟨S(Υ(s, r)), S(Υ(s, r))⟩g

(3.3.5) =∂

∂r

∣∣∣r

((⟨S, S⟩g Υ

)(s, r)

)=

(R⟨S, S⟩g

)(γr(s)).

We also note that

(3.3.6) S(γ0(s)) = γ0(s).

The length of γr is given by

(3.3.7) Lg(γr) :=∫ b

a|S(γr(s))|g ds.


Lemma 3.3.1. (First variation of arc length) Suppose 0 ∈ r. If γ0 is parametrized byarc length, that is, |S(γ0(s))|g ≡ 1, then

(3.3.8)ddr

∣∣∣r=0

Lg(γr) = −∫ b

a⟨R,∇SS⟩g (γ0(s))ds +

[⟨R, S⟩g(γ0(s))

] ∣∣∣ba.

PROOF. Calculate (use the formula (3.3.5))

ddr

Lg(γr) =12

∫ b

a|S(γr(s))|−1

g∂

∂r

∣∣∣r⟨S(γr(s)), S(γr(s))⟩g ds

=12

∫ b

a|S(γr(s))|−1

g(

R⟨S, S⟩g)(γr(s))ds.

By the assumption that |S(γ0(s))|g ≡ 1, we conclude that

ddr

∣∣∣r=0

Lg(γr) =12

∫ b

a

(R⟨S, S⟩g

)(γ0(s))ds =

∫ b

a⟨S,∇RS⟩g (γ0(s))ds.

However,

∇RS−∇SR = [R, S] = Υ∗

([∂

∂r,

∂

∂s

])= 0

which implies that

ddr

∣∣∣r=0

Lg(γr) =∫ b

a⟨S,∇SR⟩g (γ0(s))ds

Integrating by parts yields the formula (3.3.8).

Corollary 3.3.2. If γr : [0, b] → M, r ∈ r ⊂ R, is a 1-parameter family of pathsemanating from a fixed point p ∈ M (i.e., γr(0) = p) and γ0 is a geodesic parametrizedby arc length, then

(3.3.9)ddr

∣∣∣r=0

Lg(γr) =

⟨∂

∂r

∣∣∣r=0

γr(b), γ0(b)⟩

g.

Remark 3.3.3. If we do not assume γ0 is parametrized by arc length, then we have

ddr

∣∣∣r=0

Lg(γr) = −∫ b

a

⟨R,∇S

(S|S|g

)⟩g(γ0(s))ds(3.3.10)

+

[⟨R,

S|S|g

⟩g(γ0(s))

] ∣∣∣ba.

Hence, among all paths fixing two endpoints, the critical points of the length func-tional are the geodesics γ, which satisfy

∇γ

(γ

|γ|g

)= 0.


3.3.2. Second variation of arc length. Now we suppose that we have a 2-parameter family of paths γq,r : [a, b] −→ M with q ∈ q ⊂ R and r ∈ r ⊂ R.Define Φ : [a, b]× q× r→M by

(3.3.11) Φ(s, q, r) := γq,r(s).

The map Φ induces a map between tangent spaces at each point (s, q, r) ∈ [a, b]×q× r: Φ∗,(s,q,r) : T(s,q,r)([a, b]× q× r) −→ Tγq,r(s)M. We define vector fields Q, R,and S along Φ as follows:

S(γq,r(s)) = Sγq,r(s) = Φ∗,(s,q,r)

((∂

∂s

∣∣∣s

), 0, 0

)= Φ∗,(s,q,r)

(∂

∂s

∣∣∣s

),

Q(γq,r(s)) = Qγq,r(s) = Φ∗,(s,q,r)

(0,(

∂

∂s

∣∣∣q

), 0)

= Φ∗,(s,q,r)

(∂

∂s

∣∣∣q

),

R(γq,r(s)) = Rγq,r(s) = Φ∗,(s,q,r)

(0, 0,

(∂

∂s

∣∣∣r

))= Φ∗,(s,q,r)

(∂

∂r

∣∣∣r

).

Then S, R, Q ∈ C∞(Φ([a, b]× q× r), TM|Φ([a,b]×q×r)).

Lemma 3.3.4. (Second variation of arc length) Suppose 0 ∈ q and 0 ∈ r. If γ0,0 isparametrized by arc length, then

∂2

∂q∂r

∣∣∣(q,r)=(0,0)

Lg(γq,r) =∫ b

a

(⟨∇SQ,∇SR⟩g − ⟨∇SQ, S⟩g ⟨∇SR, S⟩g

)(γ0,0(s))ds

(3.3.12) −∫ b

a

⟨Rmg(Q, S)S, R

⟩g (γ0,0(s))ds

−∫ b

a

⟨∇QR,∇SS

⟩g (γ0,0(s))ds +

[⟨∇QR, S

⟩g (γ0,0(s))

] ∣∣∣ba.

PROOF. Differentiating the first variation of arc length we have

∂2

∂q∂r

∣∣∣(q,r)=(0,0)

Lg(γq,r) =∂

∂q

∣∣∣(q,r)=(0,0)

∫ b

a

⟨S|S|g

,∇SR⟩

g(γq,r(s))ds

=∂

∂q

∣∣∣(q,r)=(0,0)

∫ b

a

(Q⟨

S|S|g

,∇SR⟩

g

)(γq,r(s))ds

=∫ b

a

(⟨S|S|g

,∇Q∇SR⟩

g+

⟨∇Q

(S|S|g

),∇SR

⟩g

)(γq,r(s))ds

∣∣∣(q,r)=(0,0)

=∫ b

a

⟨S,∇S∇QR + Rmg(Q, S)R

⟩g (γ0,0(s))ds

+∫ b

a

⟨∇QS−

⟨S,∇QS

⟩g S,∇SR

⟩g(γ0,0(s))ds

where we use the identity that

(3.3.13) ∇Q

(S|S|g

)= |S|−1

g ∇QS− |S|−3g⟨S,∇QS

⟩g S.

Then the result follows from an integration by parts.


Corollary 3.3.5. If γr is a 1-parameter family of piecewise smooth paths with fixed end-points and such that γ0 is a geodesic parametrized by arc length, then

(3.3.14)d2

dr2

∣∣∣r=0

Lg(γr) =∫ b

a

(∣∣∣(∇SR)⊥∣∣∣2g−⟨Rmg(R, S)S, R

⟩g

)(γ0(s))ds,

where (∇SR)⊥ is the projection of ∇SR onto S⊥, i.e., (∇SR)⊥ := ∇SR− ⟨∇SR, S⟩g S.

PROOF. It suffices to show that∣∣∣(∇SR)⊥∣∣∣2g=⟨∇SQ, (∇SR)⊥

⟩g

which is equivalent to prove⟨⟨∇SR, S⟩g S, (∇SR)⊥

⟩g= 0.

By the definition, the left side of above equals⟨⟨∇SR, S⟩g S,∇SR− ⟨∇SR, S⟩g S

⟩g=∣∣∣⟨∇SR, S⟩g

∣∣∣2g−∣∣∣⟨∇SR, S⟩g

∣∣∣2g|S|2g = 0

since γ0 is parametrized by arc length.

A geodesic is stable if the second variation of arc length, with respect to vari-ation vector fields which vanish at the endpoints, is nonnegative.

Corollary 3.3.6. If, in addition, (M, g) has nonnegative sectional curvature and thepaths γr are smooth and closed, then

d2

dr2

∣∣∣r=0

Lg(γr) ≥ 0.

That is, any smooth closed geodesic γ0 is stable.

Theorem 3.3.7. (Synge) If (M, g) is an even-dimensional, orientable, closed Riemann-ian manifold with positive sectional curvature, thenM is simply-connected.

If γr : [0, b] → M is a 1-parameter family of paths, r ∈ (−ϵ, ϵ), γ0 is a unitspeed geodesic, and R(γ0(0)) = 0, then

d2

dr2

∣∣∣r=0

Lg(γr)− ⟨∇RR, S⟩g (γ0(b))

=∫ b

0

(∣∣∣(∇SR)⊥∣∣∣2g−⟨Rmg(R, S)S, R

⟩g

)(γ0(s))ds.(3.3.15)

Given a V ∈ Tγ0(b)M, we extend V along γ0 by defining

(3.3.16) V(γ0(s)) :=bs

R(γ0(s)),


where V(γ0(s)) is the parallel translation of V along γ0, i.e.,(∇SV

)(γ0(s)) = 0.

Note that V = V(γ0(b)) = R(γ0(b)). Then

(∇SR) (γ0(s)) = ∇S(γ0(s))

( sb

V(γ0(s)))=

1b

V(γ0(s))

so that ∇SR = 1b V. Since V(γ0(s)) is the parallel translation of V along γ0, it

follows that ∣∣∣(∇SR)⊥∣∣∣2g=

1b2

∣∣∣V⊥∣∣∣2g=

1b2

∣∣∣V⊥∣∣∣2g

where V⊥ := V − ⟨V, S⟩gS and

V⊥ := V − ⟨V, S(γ0(b))⟩gS(γ0(b)) = V − ⟨V, γ0(b)⟩gγ0(b).

Hence ∫ b

0

∣∣∣(∇SR)⊥∣∣∣2g(γ0(s))ds =

∫ b

0

1b2

∣∣∣V⊥∣∣∣2g

ds =1b

∣∣∣V⊥∣∣∣2g

and

d2

dr2

∣∣∣r=0

Lg(γr)− ⟨∇RR, S⟩g (γ0(b))

=1b

∣∣∣V⊥∣∣∣2g−∫ b

0

⟨Rmg(R, S)S, R

⟩g (γ0(s))ds.(3.3.17)

Lemma 3.3.8. If γr : [0, b]→M, r ∈ (−ϵ, ϵ), is a 1-parameter family of paths emanat-ing from a fixed point p ∈ M, i.e., γr(0) = p and γ0 is a geodesic parametrized by arclength, then

(3.3.18) dg(p, βV(r)) ≤ Lg(γr), dg(p, βV(0)) = Lg(γ0)

where βV : (−ϵ, ϵ) →M where βV(r) := γr(b) so that βV(0) = V ∈ Tγ0(b)M. Thatis, the function r 7→ dg(p, βV(r)) is a lower support function for the function r 7→ Lg(γr)at r = 0.

PROOF. According to Corollary 3.3.2, we have

ddr

∣∣∣r=0

Lg(γr) =

⟨∂

∂r

∣∣∣(s,r)=(b,0)

γr(s),∂

∂s

∣∣∣(s,r)=(b,0)

γr(s)⟩

g(γ0(b))= 0.

Hence dg(p, β(0)) = Lg(γ0).

Suppose that u ∈ C0(M) and V ∈ TpM. Let βV : (−ϵ, ϵ) → M be theconstant speed geodesic with βV(0) = p and βV(0) = V. If v : (−ϵ, ϵ) → R is aC2-function such that

u(βV(r)) ≤ v(r), r ∈ (−ϵ, ϵ), u(βV(0)) = v(0),

then we say that

(3.3.19) ∇V∇Vu ≤ v′′(0)

in the sense of support functions with respect to p and V. If (3.3.19) holds for allp and V, then we say (3.3.19) in the sense of support functions.


Exercise 3.3.9. Show that if u : M→ R satisfies ∇2u ≤ 0 in the sense of supportfunctions, then u is concave; that is, for every unit speed geodesic βV : [a, b]→Mwe have

u (βV((1− s)a + sb)) ≥ (1− s)u(βV(a)) + su(βV(b))for all s ∈ [0, 1].

We assume βV is a geodesic so that ∇QQ = 0. If the sectional curvatures arenonnegative and dg,p(x) := dg(p, x) is the distance function, then

(3.3.20) ∇V∇Vdg,p ≤d2

dr2

∣∣∣r=0

Lg(γr) ≤(

1dg,p|V|2g

)(γ0(b))

in the sense of support functions, since

b =∫ b

0|S(γ0(s))|g ds = Lg(γ0) = dg(γ0(b), p) = dg,p(γ0(b))

and

⟨V, S(γ0(b))⟩g =

⟨∂

∂r

∣∣∣r=0

γr(b),∂

∂s

∣∣∣s=b

γ0(s)⟩

g

=

⟨∂

∂r

∣∣∣(s,r)=(b,0)

γr(s),∂

∂s

∣∣∣(s,r)=(b,0)

γr(s)⟩

g= 0.

The inequality (3.3.20) is a special case (K = 0) of the Hessian comparisontheorem. Note that this inequality holds in the usual C2-sense at points where dg,pare smooth.

Lemma 3.3.10. Assuming the sectional curvature is nonnegative, one has

(3.3.21) ∇V∇Vd2g,p ≤ 2|V|2g.

PROOF. Calculate

d2

dr2

∣∣∣r=0

Lg(γr)2 =

ddr

∣∣∣r=0

(2Lg(γr) ·

ddr

Lg(γr)

)

= 2(

ddr

∣∣∣r=0

Lg(γr)

)2+ 2Lg(γr) ·

d2

dr2

∣∣∣r=0

Lg(γr)

≤ 2(

ddr

∣∣∣r=0

Lg(γr)

)2+ 2b · 1

b

∣∣∣V⊥∣∣∣2g(γ0(b))

= 2∣∣∣V⊥∣∣∣2

g(γ0(b))+ 2

(ddr

∣∣∣r=0

Lg(γr)

)2= 2

∣∣∣V⊥∣∣∣2g

= 2|V|2g.

Therefore, ∇V∇Vd2g,p ≤ 2|V|2g. Equivalently, ∇2d2

g,p ≤ 2g(γ0(b)) for any V ∈Tγ0(b)M.


3.3.3. Long stable geodesics. Let γ : [0, s] → M be a stable unit speed geo-desic in an n-dimensional Riemannian manifold (M, g) with Ricg ≤ (n− 1)Kg inBg(γ(0), r) and Bg(γ(s), r) where K > 0 and 2r < s.

Let (Ei)1≤i≤n−1 be a parallel orthonormal frame along γ perpendicular to γ.By the second variation of arc length, we have

0 ≤ ∑1≤i≤n−1

∫ s

0

(∣∣∣(∇γ(φEi(γ)))⊥∣∣∣2g−⟨Rmg(φEi(γ), γ)γ, φEi(γ)

⟩g

)ds

=∫ s

0

[(n− 1)

(dφ

ds

)2− φ2Ricg(γ, γ)

]ds

for any function φ : [0, s]→ R. Consider the piecewise smooth function

(3.3.22) φ(s) :=

sr , 0 ≤ s ≤ r,1, r < s < s− r,s−s

r , s− r ≤ s ≤ s.

We then have∫ s

0Ricg(γ, γ)ds ≤ 2(n− 1)

r+∫ s

0(1− φ2)Ricg(γ, γ)ds

=2(n− 1)

r+∫ r

0(1− φ2)Ricg(γ, γ)ds +

∫ s

s−r(1− φ2)Ricg(γ, γ)ds

≤ 2(n− 1)r

+ (n− 1)K · 4r3≤ 2(n− 1)

(1r+ Kr

).

Proposition 3.3.11. If γ : [0, L] → M is a stable unit speed geodesic in a Riemanniann-manifold with

Ricg ≤ (n− 1)Kg, in Bg

(γ(0), 1/

√K)∪ Bg

(γ(L), 1/

√K)

,

where K > 0, then ∫ L

0Ricg(γ, γ)ds ≤ 4(n− 1)

√K.

3.3.4. Jacobi fields in relation to the index form. Let γr : [a, b]→M, r ∈ r ⊂R, be a 1-parameter family of paths. Assume γ0 is a geodesic. Then S(γ0(s)) =γ0(s), and hence

(∇SS) (γ0(s)) = 0.

For the variation vector field R, we have

0 = ∇R∇SS = ∇S∇RS + Rmg(R, S)S = ∇S∇SR + Rmg(R, S)S.

Thus∇γ0(s)∇γ0(s)R(γ0(s)) + Rmg (R(γ0(s)), γ0(s)) γ0(s) = 0.

A Jaboci field J is a variation of geodesic and satisfies the Jabobi equation

(3.3.23) ∇S∇S J + Rmg(J, S)S = 0.


Given p ∈ M and V, W ∈ TpM, we define a 1-parameter family of geodesicsγr : [0, ∞)→M by

(3.3.24) γr(s) := expp (s(V + rW)) = γV+rW(s).

We may define a Jacobi field JV,W along γ0 = γV by

(3.3.25) JV,W(s) :=∂

∂r

∣∣∣r=0

γV+rW(s).

A point x ∈ M is a conjugate point of p ∈ M if x is a singular value ofexpp : TpM → M. That is, x = expp(V), for some V ∈ TpM, where (expp)∗,V :TV(TpM)→ Texpp(V)M is singular (i.e., has nontrivial kernel).

(1) Equivalently, γ(r) is a conjugate point to p along γ if there is a nontrivialJacobi field along γ vanishing at the endpoints.

(2) Given a geodesic γ : [0, L] → M without conjugate points and vectorsA ∈ Tγ(0)M and B ∈ Tγ(L)M with ⟨A, S⟩g = ⟨B, S⟩g = 0, there exists aunique Jacobi field J with J(0) = A and J(L) = B.

If γ : [a, b] → M is a path and V and W are vector fields along γ, we definethe index form of V and W by

Ig,γ(V, W) :=∫ b

a

(⟨∇SV,∇SW⟩g − ⟨∇SV, S⟩g ⟨∇SW, S⟩g

−⟨Rmg(V, S)S, W

⟩g

)ds,(3.3.26)

where S := ∂∂s |s=0γs and γs is a 1-parameter family of γ.

If γq,r is a 1-parameter family of paths with fixed endpoints and if γ0,0 is a unitspeed geodesic, then by Lemma 3.3.4,

∂2

∂q∂r

∣∣∣(q,r)=(0,0)

Lg(γq,r) = Ig,γ0,0(Q, R).

Lemma 3.3.12. (Index lemma) Suppose γ : [0, L]→M is a geodesic without conjugatepoints. In the space VectA,B(γ) of vector fields X along γ with ⟨X, S⟩g ≡ 0, X(0) = Aand X(L) = B, the Jacobi field minimizes the (modified) index form:

(3.3.27) Ig,γ(X) :=∫ L

0

(|∇SX|2g −

⟨Rmg(X, S)S, X

⟩g

)ds.

PROOF. If X and Y are vector field along γ, then

Ig,γ(X + tY) =∫ L

0

(|∇S(X + tY)|2g −

⟨Rmg(X + tY, S)S, X + tY

⟩g

)ds

=∫ L

0

(|∇SX + t∇SY|2g −

⟨Rmg(X, S)S, X

⟩g

)ds

−∫ L

0

(2t⟨Rmg(X, S)S, Y

⟩g + t2 ⟨Rmg(Y, S)S, Y

⟩g

)ds


so that12

ddt

∣∣∣t=0Ig,γ(X + tY) =

∫ L

0

(⟨∇SX,∇SY⟩g −

⟨Rmg(X, S)S, Y

⟩g

)ds.

(Note that the tangent space TXVectA,B(γ) is the space of all vector fields along γwhich vanish at the endpoints) If furthermore Y satisfies Y(0) = Y(L) = 0 , using

dds⟨Y,∇SX⟩g = ⟨∇SY,∇SX⟩g + ⟨Y,∇S∇SX⟩g ,

we obtain12

ddt

∣∣∣t=0Ig,γ(X + tY) = −

∫ L

0

⟨∇S∇SX + Rmg(X, S)S, Y

⟩g ds

and12

d2

dt2

∣∣∣t=0Ig,γ(X) =

∫ L

0

(|∇SY|2g −

⟨Rmg(Y, S)S, Y

⟩g

)ds.

Hence the critical points of Ig,γ on VectA,B(γ) are the Jacobi fields.We claim that ∫ L

0

(|∇SY|2g −

⟨Rmg(Y, S)S, Y

⟩g

)ds > 0

for any nonzero vector field Y ∈ TXVectA,B(γ), so that the index form Ig,γ is con-vex. Hence the Jacobi fields minimize Ig,γ in VectA,B(γ). We now give a varia-tional proof of this inequality.

Normalize the index by defining

ι(t) := inf0 =Z∈TXVectA,B(γ)

Ig,γ,t(Z)∫ t0 |Z(s)|2g ds

where

Ig,γ,t(Z) :=∫ t

0

(|∇SY|2g −

⟨Rmg(Y, S)S, Y

⟩g

)ds

for t ∈ [0, L].(i) First we have

dds|Z|g ≤ |∇SZ|g ,

⟨Rmg(Z, S)S, Z

⟩g ≤ C|Z|2g

for some constant C depending only on g. Indeed,

dds|Z|g =

dds⟨Z, Z⟩1/2

g =1|Z| ⟨∇SZ, Z⟩g ≤

1|Z| |Z|g |∇SZ|g = |∇SZ|g .

(ii) Second we have λ1([0, t]) = π2

t2 , where λ1([0, t]) is the first eigenvalue ofd2/ds2 with Dirichlet boundary conditions. Hence

ι(t) ≥ inf0 =Z∈TXVectA,B(γ)

∫ t0

((dds |Z|g

)2− C|Z|2g

)ds∫ t

0 |Z|2gds≥ π2

t2 − C.

For t ∈ (0, L] where γ[0,t] is minimizing (e.g., for t > 0 small enough), wehave ι(t) ≥ 0.


Since Ig,γ,t is a second variation of γ|[0,t] vanishing at the endpoints 0 andt, and ι(t) is continuous, if the claim is not ture, we can find t0 ∈ (0, L] such thatι(t0) = 0. Then Ig,γ,t0(Z0) = 0 for some vector field Z0 with Z0(0) = 0, Z0(t0) = 0,and Z0 = 0. By considering the Euler-Lagrange equation for

E(Z) :=Ig,γ,t0(Z)∫ t0

0 |Z(s)|2g ds

at Z0, we have for all W vanishing at 0 and t0,

0 =12

ddu

∣∣∣u=0

E(Z0 + uW)

= − 1∫ t00 |Z0(s)|2g ds

∫ t0

0

⟨∇S∇SZ0 + Rmg(Z0, S)S, W

⟩g ds,

since Ig,γ,t0(Z0) = 0. Thus Z0 is a nontrivial Jacobi field along γ|[0,t0]with Z0(0) =

0 = Z0(t0). This contradicts the assumption that there are no conjugate pointsalong γ. Hence ι(t) > 0 for all t ∈ (0, L].

3.3.5. First and second variation of energy. Given a path γ : [a, b] → M, itsenergy is defined by

(3.3.28) Eg(γ) :=12

∫ b

a|γ(s)|2g ds.

Let γr : [a, b] → M denote a 1-parameter family of paths, r ∈ r ∈ R. We also usethe variation vector field R and the tangent vector field S. The length of γr is givenby

Eg(γr) :=12

∫ b

a|γr(s)|2g ds.

Lemma 3.3.13. (First variation of energy) Suppose 0 ∈ r. The first variation of energyis

(3.3.29)ddr

∣∣∣r=0

Eg(γr) = −∫ b

a

(⟨R,∇SS⟩g

)(γ0(s))ds +

[⟨R, S⟩g(γ0(s))

] ∣∣∣ba.

PROOF. Calculateddr

∣∣∣r=0

Eg(γr) =12

∫ b

a

∂

∂r

∣∣∣r=0⟨γr(s), γr(s)⟩g (γr(s))ds

=12

∫ b

a

(R⟨S, S⟩g

)(γ0(s))ds =

∫ b

a

(⟨∇RS, S⟩g

)(γ0(s))ds

=∫ b

a

(⟨∇SR, S⟩g

)(γ0(s))ds

that implies the lemma.

The critical points of the energy, among all paths fixing two endpoints, are theconstant speed geodesics γ, which satisfy

∇γγ = 0.


The speed of γ is constant since γ|γ|2g = 2 ⟨∇γγ, γ⟩g = 0.

Let γq,r : [a, b] → M with q ∈ q ⊂ R and r ∈ r ⊂ R, be a 2-parameter familyof paths. Recall the definition of vector fields Q, R, and S.

Lemma 3.3.14. (Second variation of energy) Suppose 0 ∈ q and 0 ∈ r. Then thesecond variation of energy is

∂2

∂q∂r

∣∣∣(q,r)=(0,0)

Eg(γq,r) =∫ b

a

(⟨∇SQ,∇SR⟩g

)(γ0,0(s))ds

+∫ b

a

⟨Rmg(Q, S)R, S

⟩g (γ0,0(s))ds(3.3.30)

−∫ b

a

⟨∇QR,∇SS

⟩g (γ0,0(s))ds +

[⟨∇QR, S

⟩g

](γ0,0(s))

∣∣∣ba.

PROOF. Since∂2

∂q∂r

∣∣∣(q,r)=(0,0)

Eg(γq,r) =∫ b

a

(Q ⟨S,∇SR⟩g

)(γ0,0(s))ds

=∫ b

a

(⟨∇QS,∇SR

⟩g +

⟨S,∇Q∇SR

⟩g

)(γ0,0(s))ds

we prove the lemma.

3.3.6. First and second variation of area. Let xr : Sn−1 →M be a parametrizedhypersurface in an n-dimensional Riemannian manifold (M, g) evolving by

(3.3.31) ∂rxr = βrνr

where βr is some function on Sn−1r := xr(Sn−1). In terms of local coordinates

(xi)1≤i≤n−1 on Sn−1, the area element of Sn−1r is

(3.3.32) dS′V =√

det(gij)dx1 ∧ · · · ∧ dxn−1.

Then

(3.3.33) ∂rgij = 2βrhij.

Hence

(3.3.34) ∂rdV′r =12

gij (∂rgij)

dV′r = βr HrdV′r .

Thus the first variation of

(3.3.35) Ag(Sn−1r ) :=

∫Sn−1

r

dV′r

is

(3.3.36)ddr

Ag(Sn−1r ) =

∫Sn−1

r

βr Hr dV′r .

Under the hypersurface flow (3.3.31), we have

(3.3.37) ∂r Hr = −∆gr βr − |hr|2gr βr − Ricgr (νr, νr)βr.

3.4. EXPONENTIAL MAPS AND NORMAL COORDINATES 171

When βr = −Hr, the mean curvature flow, we have

(3.3.38) ∂r Hr = ∆gr Hr + |hr|2gr Hr + Ricg (νr, νr) Hr.

Now, we can compute the second variation of area:

d2

dr2 Ag(Sn−1r ) =

∫Sn−1

r

βr

(−∆gr βr − |hr|2gr βr − Ricg (νr, νr) βr + H2

r βr

)dS′r

=∫Sn−1

r

(∣∣∇gr βr∣∣2gr+(

H2r − |hr|2gr − Ricg (νr, νr)

)β2

r

)dS′r.

If ∂rxr = νr, then

d2

dr2 Ag(Sn−1r ) = −

∫Sn−1

r

(H2

r − |hr|2gr − Ricg(νr, νr))

dS′r.

However,

(3.3.39) Rgr = Rg − 2Ricg(νr, νr) + H2r − |hr|2gr .

Therefore, if ∂rxr = νr, then the second variation of area is given by

(3.3.40)d2

dr2 Ag(Sn−1r ) =

12

∫Sn−1

r

(Rgr − Rg + H2

r − |hr|2gr

)dS′r.

Theorem 3.3.15. (Schoen-Yau, 1979) If S2 is an orientable closed stable minimal surfacein a 3-manifold (M3, g) with positive scalar curvature, then S2 is diffeomorphic to a 2-sphere.

PROOF. Let S2r be a variation of S2 with S2

0 = S2 and β = 1, by (3.3.35),H ≡ 0, and the Gauss-Bonnet formula, we have

0 ≤ d2

dr2

∣∣∣r=0

Ag(S2r ) =

12

∫S2

(RS2 − RM3 − |h|2g

)dV′g

= 2πχ(S2)− 12

∫S2

(RM3 + |h|2g

)dV′g.

Since RM3 > 0 and |h|2g ≥ 0, it follows that χ(S2) > 0. Since M3 is orientable,S2 ∼= S2.

3.4. Exponential maps and normal coordinates

Let (M, g) be an n-dimensional Riemannian manifold and p ∈ M. For V ∈TpM, there is a unique constant speed geodesic γV : [0, bV) →M is the constantspeed geodesic emanating from p with γV(0) = V. Here [0, bV) is the maximaltime interval on which γV is defined.


3.4.1. Exponential map. For all α > 0 and t < bαV , we have

(3.4.1) γαV(t) = γV(αt), bαV = α−1bV .

Let Op ⊂ TpM be the set of vectors V such that 1 < bV , so that γV(t) is definedon [0, 1]. Then define the exponential map at p by

(3.4.2) expp : Op −→M, V 7−→ γV(1).

If bV > t, then btV = t−1bV > 1 and

(3.4.3) expp(tV) = γtV(1) = γV(t).

Remark 3.4.1. If M is compact, then for each p ∈ M and V ∈ TpM, there is aunique constant speed geodesic γV : [0, ∞)→M with γ(0) = p and γV(0) = V.

Let O := ∪p∈MOp. Then the exponential map expp induces a map

(3.4.4) exp : O −→M

by setting exp|Op= expp. This map is also called the exponential map. Fur-

thermore, the set O is open in TM and exp : O → M is smooth. In addition,Op ⊂ TpM is open and expp : Op →M is also smooth.

Proposition 3.4.2. (1) If p ∈ M, then

(3.4.5) d expp : T0(TpM) −→ TpM

is nonsingular at the origin of TpM. Consequently expp is a local diffeomorphsm.(2) Define Exp : O→M×M by

Exp(V) =(

π(V), expπ(V) V)

,

where π(V) is the base point of V, i.e., V ∈ Tπ(V)M. Then for each p ∈ M and with itthe zero vector, 0p ∈ TpM,

dExp(p,0p): T(p,0p)(TM) −→ T(p,p)(M×M)

is nonsingular. Consequently, Exp is a diffeomorphism from a neighborhood of the zerosection of TM onto an open neighborhood of the diagonal inM×M.

PROOF. Let I0 : TpM→ T0(TpM) be the canonical isomorphism, i.e.,

(3.4.6) I0(V) :=ddt(tV)

∣∣∣t=0

.

Recall that if V ∈ Op, then γV(t) = γtV(1) for all t ∈ [0, 1]. Thus,

d expp(I0(V)) =ddt

∣∣∣t=0

expp(tV) =ddt

∣∣∣t=0

γtV(1)

=ddt

∣∣∣t=0

γV(t) = γV(0) = V.


In other words, d expp I0 is the identity map on TpM. This shows that d expp isnonsingular.

For (2), we note that the tangent space T(p,p)(M×M) is naturally identifiedwith TpM× TpM. The tangent space T(p,0p)(TM) is also naturally identifiedto Tp M × T0p(TM) ∼= TpM× TpM. We know that d Exp(p,0p)

takes (p, V) to(p, expp(V)). Under above identification, if we consider the map dExp(p,0p)

as alinear map TpM× TpM→ TpM× TpM, then it is the identity on the first factorto the first factor, identically 0 from the second factor to the first, and the identityfrom the second factor to the second factor as it is d expp I0p . Thus it looks like(

I 0∗ I

)which is clearly nonsingular.

Remark 3.4.3. Suppose that N is an embedded submanifold of M. The normalbundle of N inM is the vector bundle over N consisting of the orthogonal com-plements of the tangent spaces TpN ⊂ TpM:

(3.4.7) T⊥N := (p, V) : V ∈ TpM, p ∈ N , V ∈ (TpN )⊥ ⊂ TpM.So for each p ∈ N ,

TpM = TpN ⊕ T⊥p N

is an orthogonal direct sum. Define the normal exponential map exp⊥ by restrict-ing exp to O ∩ T⊥N , so

exp⊥ : O ∩ T⊥N →M.As in part (2) of Proposition 3.4.2, d exp⊥ is nonsingular at 0p, p ∈ N . Then itfollows that there is an open neighborhood U of the zero section in T⊥N on whichexp⊥ is a diffeomorphism onto its image inM. Such an image exp⊥(U ) is calleda tubular neighborhood of N inM.

Theorem 3.4.4. Suppose that (M, g) is a Riemannian manifold, p ∈ M, and ϵ > 0 issuch that

expp : B(0p, ϵ) ⊂ TpM−→ U ⊂Mis a diffeomorphism onto its image U := expp(B(0p, ϵ)) inM. Then U = Bg(p, ϵ) andfor each V ∈ B(0p, ϵ), the geodesic γV : [0, 1]→M defined by

γV(t) := expp(tV)

is the unique minimal geodesic inM from p to expp(V).

On U we have the function r(x) := | exp−1p (x)|. That is, r is the Euclidean

distance function from the origin on B(0p, ϵ) ⊂ TpM in exponential coordinates.

3.4.2. Gauss lemma and the Hopf-Rinow theorem. Let (M, g) be a Riemann-ian manifold and p ∈ M. Suppose that V ∈ TpM and for some L > 0 the constant


speed geodesic γV with γV(0) = V is defined on [0, L] for every V in some neigh-borhood of V. Given u ∈ (0, L), let

Lemma 3.4.5. (Gauss Lemma) If W ∈ TuV(TpM) ∼= TpM is perpendicular to V,then the image (expp)∗,uV(W) of W is perpendicular to (expp)∗,p(V) = γV(u):

(3.4.8)⟨(expp)∗,uV(W), (expp)∗,uV(V)

⟩g= 0.

If the distance function r := dg(p, ·) is smooth at a point x, we then have

(3.4.9) ∇gr(x) = γ0(b),

where γ0 : [0, b] → M is the unique unit speed minimal geodesic from p to x. Thus, ifγ0 = γV0 for some unit vector V0, then ∇gr = (expp)∗,0(V0).

PROOF. Given V, W ∈ TpM, we define the family of geodesics

γr(s) := expp(s(uV + rW)), 0 ≤ s ≤ 1.

Then Lg(γr) = |uV + rW|g(p) and

ddr

∣∣∣r=0

Lg(γr) =ddr

∣∣∣r=0⟨uV + rW, uV + rW⟩1/2

g(p)

=2u⟨V, W⟩g(p)

2|uV + rW|g(p)

∣∣∣r=0

=1

|V|g(p)⟨V, W⟩g(p).

On the other hand, by Lemma ??, we have

ddr

∣∣∣r=0

Lg(γr) = − 1u|V|g(p)

∫ 1

0⟨R,∇SS⟩gds +

1u|V|g(p)

⟨R, S⟩g∣∣∣10

=1

u|V|g(p)

⟨(expp)∗,uV(W), (expp)∗,uV(uV)

⟩g

=1

|V|g(p)

⟨(expp)∗,uV(W), (expp)∗,uV(V)

⟩g

since γr are geodesics. Thus⟨(expp)∗,uV(W), (expp)∗,uV(V)

⟩g= ⟨V, W⟩g = 0

which proves the first part.Let γr : [0, b] → M, r ∈ r, be an arbitrary variation of γ0 with γr(0) =

p. Since γ0 is a minimal geodesic, we have Lg(γr) ≥ dg(p, γr(b)) and Lg(γ0) =dg(p, γ0(b)). Hence, since ∇gr exists at x,

⟨∇r, X⟩g(x) =ddr

∣∣∣r=0

Lg(γr),

where X := ∂∂r |r=0γr(b). On the other hand, by the first variation formula (3.3.8),

ddr

∣∣∣r=0

Lg(γr) =

⟨∂

∂s

∣∣∣s=0

γs(b), X⟩

g= ⟨γ0(b), X⟩g.

Therefore, ∇r = γ0(b).


Let ∂/∂r denote the radial unit outward pointing vector field on TpM\ 0and consider the map, where X ∈ TpM,

(expp)∗,X : TX(TpM) −→ Texpp(X)M.

We denote by ∂X the canonical isomorphism

(3.4.10) ∂X : TpM−→ TX(TpM), Y 7−→ ∂XY :=ddt

∣∣∣t=0

(X + tY).

Hence we obtain

(3.4.11) (expp)∗,X := (expp)∗,X ∂0 : TpM−→ Texpp(X)M.

Since (expp)∗,0 = idTpM is invertible, there exists an ϵ > 0 such that expp restrictedon the punctured ball B(0, ϵ) \ 0 ⊂ TpM is an embedding. We denote

rg(x) :=∣∣∣exp−1

p (x)∣∣∣g(p)

, x ∈ Bg(p, ϵ) := expp(B(0, ϵ)).

If r, θ1, · · · , θm−1 are spherical coordinates in TpM, then we set

∂

∂rg

∣∣∣expp(X)

:= (expp)∗,X

(∂

∂r(X)

),

∂

∂θig

∣∣∣expp(X)

:= (expp)∗,X

(∂

∂θi (X)

).

For every X ∈ TxM, we may write it as

X = a∂

∂rg

∣∣∣x+

m−1

∑i=1

bi ∂

∂θig

∣∣∣x;

By Gauss lemma 3.4.8, one has⟨∂

∂rg

∣∣∣x,

∂

∂θig

∣∣∣x

⟩g

= 0

and hence ⟨∂

∂rg

∣∣∣x, X⟩

g= a = X(rg) =

⟨gradg(rg(x)), X

⟩g

.

Thus

(3.4.12)∂

∂rg

∣∣∣x= gradg(rg)(x), x ∈ Bg(p, ϵ).

Lemma 3.4.6. For every V ∈ B(0, ϵ), γV : [0, 1] → M is the unit path, up toreparametrization, joining p and γV(1) = expp(V) whose length realizes the distancedg(p, expp(V)) = |V|g. In particular, short geodesics are minimal and rg(x) = dg(p, x)for x ∈ Bg(p, ϵ).

PROOF. Since ∂/∂r is unit, it follows from (3.4.12) that

|β(u)|g ≥⟨

β(u),∂

∂rg

∣∣∣β(u)

⟩g

=⟨

β(u), gradg(rg)(β(u))⟩

g= β(u)(rg) =

ddu

rg(β(u))


for any path β from p to expp(V) that stays inside Bg(p, ϵ), so that

|V|g = rg(expp(V)) =∫ 1

0

ddu

rg(β(u))du ≤∫ 1

0|β(u)|g(β(u)) du ≤ dg(p, expp(V)).

Hence γV realizes the distance from p to expp(V).

Theorem 3.4.7. (Hopf-Rinow) Let (M, g) be a Riemannian manifold. Then the follow-ing are equivalent:

(1) (M, dg) is a complete metric space.(2) There exists p ∈ M such that expp is defined on all of TpM.(3) For all p ∈ M, expp is defined on all of TpM.

Any one of these conditions implies(4) For any p, q ∈ M there exists a smooth minimal geodesic form p to q.

3.4.3. Cut locus and injectivity radius. Let (M, g) be a Riemannian mani-fold. A function f : M → R is a globally Lipschitz function with Lipschitzconstant C if for all x, y ∈ M we have

| f (x)− f (y)| ≤ Cdg(x, y).

If for every z ∈ M there exists a neighborhood Uz of z and a constant Cz such that

| f (x)− f (y)| ≤ Cz dg(x, y)

for all x, y ∈ Uz, then we say that f is a locally Lipschitz function.

The distance function dg(p, ·) is a globally Lipschitz function with Lipschitzconstant 1.

Given a point p ∈ M and a unit speed geodesic γ : [0, ∞) →M with γ(0) =p, either γ is a geodesic ray (i.e., minimal on each finite subinterval) or there existsa unique rγ ∈ (0, ∞) such that dg(p, γ(r)) = r for r ≤ rγ and dg(p, γ(r)) < r forr > rγ. We say that γ(rγ) is a cut point to p along γ.

(i) If γ(r) is a conjugate point to p along γ, then r ≥ rγ.(ii) The cut locus Cutg(p) of p inM is the set of all cur points of p.

(iii) Let

(3.4.13) Dg(p) := V ∈ TpM : dg(p, expp(V)) = |V|g,

which is a closed subset of TpM. We define Cg(p) := ∂Dg(p) to be thecut locus of p in the tangent space. We have

(3.4.14) Cutg(p) = expp(Cg(p))

andexpp : int(Dg(p)) ⊂ TpM−→M\Cutg(p)

is a diffeomorphism. We call int(Dg(p)) the interior to the cut locus inthe tangent space TpM.


Lemma 3.4.8. A point γ(r) is a cut point to p along γ if and only if r is the smallestpositive number such that either γ(r) is a conjugate point to p along γ or there exist twodistinct minimal geodesics joining p and γ(r).

Given V ∈ TpM and r > 0, we have γV(r) = expp(rV). For each unit vectorV ∈ TpM there exists at most a unique rV ∈ (0, ∞) such that γV(rV) is a cut pointof p along γV . Furthermore, if we set rV = ∞ when γV is a ray, then the map fromthe unit tangent space at p to (0, ∞] given by V 7→ rV is a continuous function.Hence we have

(3.4.15) Cg(p) = ∂Dg(p) = rVV : V ∈ TpM, |V|g(p) = 1, γV is not a ray

has measure zero with respect to the Euclidean measure on (TpM, g(p)).

Lemma 3.4.9. Cutg(p) = expp(Cg(p)) has measure zero with respect to the Riemann-ian measure on (M, g).

If x /∈ Cutg(p) and x = p, then dg(p, ·) is smooth at x and |∇dg(p, x)|g(x) = 1by (3.4.9). Since Cutg(p) has measure zero, we have |∇dg(p, ·)|g = 1 a.e. onM.

The injectivity radius injg(p) of a point p ∈ M is defined to the supremum ofall r > 0 such that expp is an embedding when restricted to B(0, r). Equivalently,

(1) injg(p) is the distance from 0 to Cg(p) with respect to g(p).(2) injg(p) is the Riemannian distance from p to Cutg(p).

The injectivity radius of a Riemannian manifold (M, g) is defined to be

(3.4.16) injg(M) := infp∈M

injg(p).

WhenM is compact, the injectivity radius is always positive.

Theorem 3.4.10. (Klingenberg) (1) If (M, g) is a compact Riemannian manifold withSecg ≤ K, then

(3.4.17) injg(M) ≥ min

π√K

,12· length of shortest closed geodesic

.

(2) If (M, g) is a complete simply-connected Riemannian manifold with 0 < 14 K <

Secg ≤ K, then

injg(M) ≥ π√K

.

(3) If (M, g) is a compact, even-dimensional, orientable Riemannian manifold with0 < Secg ≤ K, then

injg(M) ≥ π√K

.


3.5. Second fundamental forms of geodesic spheres

In this section we consider geodesic spherical coordinates and the second fun-damental forms and mean curvatures of geodesic spheres. We also give the proofsof the Laplacian and Hessian comparison theorems for the distance function andthe corresponding volume and Rauch comparison theorems.

3.5.1. Geodesic coordinate expansion of the metric and volume form. Let(M, g) be a Riemannian manifold of dimension n. The exponential map expp :TpM → M is defined by expp(V) := γV(1), where γV : [0, ∞) → M is theconstant speed geodesic emanating from p with γV(0) = V.

Given an orthonormal frame (ei)1≤i≤n at p, let (Xi)1≤i≤n denote the standardEuclidean coordinates on TpM defined by V = ∑1≤i≤n Viei. Geodesic coordi-nates or normal coordinates are defined by

(3.5.1) xi := Xi exp−1p :M\Cutg(p) −→ R.

In geodesic coordinates, we have

gij = δij −13

Ripqjxpxq − 16∇rRipqjxpxqxr

+

(− 1

20∇r∇sRipqj +

245

guvRipquRjrsv

)xpxqxrxs + O(r5

g)(3.5.2)

so that gij = δij + O(r2g), and

det(g) = 1− 13

Rijxixj − 16∇kRijxixjxk

−(

120∇ℓ∇kRij +

190

RpijqRpkℓ

q − 118

RijRkℓ

)xixjxkxℓ + O(r5

g).(3.5.3)

Lemma 3.5.1. (Expansion for volumes of balls) One has

(3.5.4) Vol(Bg(p, r)) = ωnrn[

1−Rg(p)

6(n + 2)r2 + O(r3)

].

PROOF. It follows from√det(g)(x) = 1− 1

6Rij(p)xixj + O(r3

g(x))

by (3.5.3).

Lemma 3.5.2. In geodesic coordinates centered at a point p ∈ M we have

(3.5.5) gij(p) = δij,∂

∂xi gjk(p) = 0.

3.5. SECOND FUNDAMENTAL FORMS OF GEODESIC SPHERES 179

3.5.2. Geodesic spherical coordinates and the Jacobian. We say that the ge-ometry is bounded or controlled if there is a curvature bound and an injectivityradius lower bound.

Given a point p ∈ M, let (Xi)1≤i≤n be local spherical coordinates on TpM\p. That is,

(3.5.6) Xn(V) := r(V) = |V|g(p), Xi(V) := θi(

V|V|g

)for 1 ≤ i ≤ n− 1,

where (θi)1≤i≤n−1 are local coordinates on Sn−1p := V ∈ TpM : |V|g(p) = 1. Let

expp : TpM→M be the exponential map. We call the coordinate system

(3.5.7) x := xi := Xi exp−1p : Bg(p, injg(p)) \ p −→ Rm

a geodesic spherical coordinate system. Abusing notation, we let

(3.5.8) rg := xn, θig := xi

for i = 1, · · · , n− 1, so that

(3.5.9)∂

∂rg= (x−1)∗

∂

∂Xm ,∂

∂θig= (x−1)∗

∂

∂Xi ,

which form a basis of vector fields on Bg(p, injg(p)) \ p. Recall from (3.4.12) that

the Gauss lemma says that gradgrg = ∂∂rg

at all points outside the cut locus of p,so that

(3.5.10)∣∣∣gradgrg

∣∣∣2g=

∣∣∣∣ ∂

∂rg

∣∣∣∣2g=

⟨gradgrg,

∂

∂rg

⟩g=

∂

∂rgrg = 1

and

gin := g

(∂

∂rg,

∂

∂θig

)=

⟨∂

∂Xn ,∂

∂Xi

⟩g= 0

for i = 1, · · · , m− 1. We may then write the metric as

(3.5.11) g = drg ⊗ drg + gijdθig ⊗ dθ

jg,

where gij := g(∂/∂θig, ∂/∂θ

jg).

Along each geodesic ray emanating from p, ∂/∂θig is a Jacobi field, before the

first conjugate point for each 1 ≤ i ≤ n− 1. We call

(3.5.12) Jg :=√

det(gij)1≤i,j≤m−1

the Jacobian of the exponential map. The volume of g is

(3.5.13) dVg =√

det(g)dθ1g ∧ · · · ∧ dθm−1

g ∧ drg = Jg dΘg ∧ drg

in a positively oriented spherical coordinate system, where

(3.5.14) dΘg := dθ1g ∧ · · · ∧ dθn−1

g .

Hence the Jacobian of the exponential map is the volume density in spherical co-ordinates. If γ(r) is a conjugate point to p along γ, then Jg(γ(r))→ 0 as r → r.


Remark 3.5.3. Along a geodesic ray γ emanating from p we have that

(3.5.15) limx→p

((∇g) ∂

∂rg

∂

∂θig

)(γ(x)) := Ei ∈ TpM

exists. Suppose (Ei)1≤i≤n−1 is orthonormal (one can always choose such geodesicspherical coordinates and we shall often make this assumption in the sequel).Then

(3.5.16) limrg→0

Jg(γ(rg))

rm−1g

= 1.

Intuitively, one way to see that (3.5.16) holds is to note that (M, cg, p) convergesas c → ∞ in the pointed limit (Rn, 0), so that the limit in (3.5.16) should equal theEuclidean value.

3.5.3. The second fundamental form of distance spheres and the Ricatti equa-tion. Consider the distance spheres

(3.5.17) Sg(p, r) := x ∈ M : dg(p, x) = r.

Let h denote the second fundamental form of Sg(p, r). We have

hij := h

(∂

∂θig

,∂

∂θjg

)=

⟨(∇g) ∂

∂θig

∂

∂rg,

∂

∂θjg

⟩g

= −⟨

∂

∂rg, (∇g) ∂

∂θig

∂

∂θjg

⟩g

= −Γnij =

12

∂

∂rggij(3.5.18)

since ∂/∂rg is the unit normal to Sg(p, r) and gin = gjn = 0. The mean curvatureH of Sg(p, r) is

(3.5.19) H = −gijΓnij =

12

gij ∂

∂rggij =

∂

∂rgln Jg.

Remark 3.5.4. (1) For rg small enough,

hij =1rg

gij + O(rg),(3.5.20)

H =n− 1

rg+ O(rg).(3.5.21)

(2) In spherical coordinates, the Laplacian is

∆g = gab(

∂2

∂xa∂xb − Γcab

∂

∂xc

)=

∂2

∂r2g+ H

∂

∂rg+ ∆Sg(p,r)(3.5.22)

=∂2

∂r2g+

(∂

∂rgln√

det(g))

∂

∂rg+ ∆Sg(p,r)


since Γann = 0 for a = 1, · · · , m and where ∆Sg(p,r) is the Laplacian with respect to

the induced metric on Sg(p, r).

Lemma 3.5.5. We have the Ricatti equation

(3.5.23)∂

∂rhij = −Rnijn + hikgkℓhℓj

where Rnmijn := ⟨Rmg(∂

∂rg, ∂

∂θig) ∂

∂θjg, ∂

∂rg⟩g. In particular,

(3.5.24)∂

∂rgH = −Ricg

(∂

∂rg,

∂

∂rg

)− |h|2g.

PROOF. Since |∂/∂rg|g = 1, it follows that ∇∂/∂rg(∂/∂rg) = 0. From (3.5.18),we have

∂

∂rghij = − ∂

∂rg

⟨∂

∂rg,∇ ∂

∂θig

∂

∂θjg

⟩g

= −⟨

∂

∂rg,∇ ∂

∂rg∇ ∂

∂θig

∂

∂θjg

⟩g

= −⟨

∂

∂rg,∇ ∂

∂θig

∇ ∂∂rg

∂

∂θjg

⟩g

− Rnijn

= − ∂

∂θig

⟨∂

∂rg,∇ ∂

∂rg

∂

∂θjg

⟩g

+

⟨∇ ∂

∂θig

∂

∂rg,∇ ∂

∂rg

∂

∂θjg

⟩g

− Rnijn

= − ∂

∂θig

∂

∂rg

⟨∂

∂rg,

∂

∂θig

⟩g

−⟨∇ ∂

∂rg

∂

∂rg,

∂

∂θig

⟩g

+

⟨∇ ∂

∂θig

∂

∂rg,∇ ∂

∂θjG

∂

∂rg

⟩g

− Rnijn = 0 + hikhjℓgkℓ − Rnijn.

Since∂

∂rgH = gij ∂

∂rghij −

∂

∂rggij · hij,

∂

∂rggij = hij,

we obtain (3.5.24).

Remark 3.5.6. If Rcg ≥ (m− 1)K, then

(3.5.25)∂

∂rg

(H

m− 1

)≤ −K−

(H

m− 1

)2.

From (3.5.21), one has

limrg→0

rgHm− 1

= 1.


In terms of the radial covariant derivative

∇nhij :=(∇ ∂

∂rgh)

ij=

∂

∂rghij − Γk

nihkj − Γknjhik

and Γkni = hi

k, we deduce from (3.5.23) that

(3.5.26) ∇nhij = −Rnijn − hikhjℓgkℓ.

Invariantly, we write this as

(3.5.27)(∇ ∂

∂rg

)(X, Y) = −

⟨Rmg

(∂

∂rg, X)

Y,∂

∂rg

⟩g− h2(X, Y)

for X, Y ∈ TSg(p, r).

3.5.4. Space form and rotationally symmetric metrics. We consider the geo-desic spheres in simply-connected space form (MK, gK). In this case the metric isgiven by

(3.5.28) gK := dr2g + s2

K(rg)gSm−1 ,

where

(3.5.29) sK(rg) :=

1√K

sin(√

Krg), K > 0,rg, K = 0,

1√K

sinh(√|K|rg), K < 0.

Lemma 3.5.7. (Curvatures of a rotationally symmetric metric) If

(3.5.30) g = dr2g + ϕ2(rg)gSm−1

for some function ϕ, which is called a rotationally symmetric metric, then the sectionalcurvatures are

(3.5.31) Krad = −ϕ′′

ϕ, Ksph =

1− (ϕ′)2

ϕ2 ,

where Krad (rad for radial) or Ksph (sph for spherical) is the sectional curvature of planescontaining or perpendicular to, respectively, the radial vector. As a consequence, we have

(3.5.32) Rcg = −(n− 1)ϕ′′

ϕdr2

g +[(n− 2)

(1− (ϕ′)2

)− ϕ′′ϕ

]gSn−1

and

(3.5.33) Rg = −2(n− 1)ϕ′′

ϕ+ (n− 1)(n− 2)

1− (ϕ′)2

ϕ2 .

Furthermore, the Laplacian of g is

(3.5.34) ∆g =∂2

∂r2g+ (n− 1)

ϕ′

ϕ

∂

∂rg+ ∆Sg(p,r).

PROOF. One way is to use the Cartan structure equations: ωn = drg and ωi =

ϕ(rg)ηi, where (ηi)1≤i≤n−1 is a local orthonormal coframe field for (Sn−1, gSn−1).


Another way of deriving (3.5.31) is to consider the distance spheres. From (3.5.18),we have

(3.5.35) hij =12

∂

∂rggij = ϕϕ′gSn−1 =

ϕ′

ϕgij, i, j = 1, ·, n− 1.

That is, the distance spheres are totally umbillic with principal curvatures κ equalto ϕ′/ϕ. The intrinsic curvature of the hypersurface Sg(p, r) is

(3.5.36) Kin =1

ϕ2 .

From the Gauss equations, we have Ksph = Kin − κ2.

Using (3.5.19) yields

(3.5.37) H = gijhij = (m− 1)ϕ′

ϕ.

Example 3.5.8. When ϕ(rg) = sK(rg) given by (3.5.29), the mean curvature HK(rg)of the distance sphere SK(p, r) is

(3.5.38) HK(rg) :=

(n− 1)

√Kcot(

√Krg), K > 0,

n−1rg

, K = 0,

(n− 1)√|K|coth(

√|K|rg), K < 0.

Note that HK(rg) is a solution to the equality case of (3.5.25). That is,

(3.5.39)∂

∂rg

(HK(rg)

n− 1

)= −K−

(HK(rg)

n− 1

)2

and

limrg→0+

rg HK(rg)

n− 1= 1.

3.5.5. Mean curvature of geodesic spheres and the Bonnet-Myers theorem.By the ODE comparison theorem, we have

Lemma 3.5.9. (Mean curvature of distance spheres comparison) If the Ricci cur-vature of (M, g) satisfies the lower bound Ricg ≥ (n− 1)Kg for some K ∈ R, then themean curvatures of the distance spheres Sg(p, r) satisfy

(3.5.40) H ≤ HK

at points where the distance function is smooth.

PROOF. From (3.5.25) and (3.5.39), we have

∂

∂rg(H − HK) ≤ −

H + HKn− 1

(H − HK).


Note that (H− HK)(rg) = O(rg). Integrating (3.5.40), we get that for any rg ≥ ϵ >0,

(3.5.41) (H − HK)(rg) ≤ (H − HK)(ϵ) · exp[−∫ rg

ϵ

H + HKn− 1

(s)ds]

.

Letting ϵ→ 0 yields (H − HK)(rg) ≤ 0.

Theorem 3.5.10. (Bonnet-Myers) If (M, g) is a complete Riemannian manifold withRicg ≥ (n − 1)Kg, where K > 0, then diam(M, g) ≤ π/

√K. In particular, M is

compact and π1(M) < ∞.

PROOF. Consider any point p ∈ M and suppose γ : [0, L] → M is a unitspeed minimal geodesic emanating from p. Then dg(p, ·) is smooth on γ((0, L))and for every r ∈ (0, L), the distance sphere Sg(p, r) is smooth in a neighborhoodof γ(r). By Lemma 3.5.9, we have

H(rg) ≤ (n− 1)√

K cot(√

Krg)

along γ|(0,L). Since

limrg→(π/

√K)+

cot(√

Krg) = −∞,

it forces that L ≤ π/√

K. Thus diam(M, g) ≤ π/√

K. Now a complete Riemann-ian manifold with finite diameter is compact.

Furthermore, we may apply the diameter bound to the universal coveringRiemannian manifold (Mn, g), where g is the lifted metric. Indeed, g satisfiesthe same Ricci curvature lower bound as g. This implies Mn is compact and weconclude π1(M) < ∞.

3.6. Comparison theorems

Two fundamental results in Riemannian geometry are the Laplacian and Hes-sian comparison theorems for the distance function. They are directly related tothe volume comparison theorem and a special case of the Rauch comparison theo-rem. The Hessian comparison theorem may also be used to prove the Toponogovtriangle comparison theorem.

3.6.1. Laplacian comparison theorem. The idea of comparison theorem is tocompare a geometric quantity on a Riemannian manifold with the correspondingquantity on a model space.

Theorem 3.6.1. (Laplacian comparison) If (M, g) is a complete Riemannian manifoldwith Ricg ≥ (n − 1)Kg, where K ∈ R, and if p ∈ M, then for any x ∈ M wheredg(x) := dg(p, x) is smooth, we have

(3.6.1) ∆dg(x) ≤

(n− 1)

√K cot

(√Kdg(x)

), K > 0,

n−1dg(x) , K = 0,

(n− 1)√|K| coth

(√|K|dg(x)

), K < 0.

3.6. COMPARISON THEOREMS 185

On the whole manifold, the Laplacian comparison theorem (3.6.1) holds in the sense ofdistributions.

In general, we say that ∆g f ≤ F in the sense of distributions if for any non-negative C∞-function φ onM with compact support, we have∫

Mf ∆φdVg ≤

∫M

FφdVg.

PROOF. If rg(x) := dg(x) is the distance function to p, then since rg is constanton each sphere ∆Sg(p,r) = 0, then from (3.5.22) we have that the Laplacian of thedistance function is the radial derivative of the logarithm of the Jacobian (and isthe mean curvature of the distance spheres)

(3.6.2) ∆rg = H =∂

∂rln Jg.

Hence, if Ricg ≥ (n− 1)Kg, then, by Lemma 3.5.9,

(3.6.3) ∆rg ≤ HK(rg).

This proves the Laplacian comparison theorem assuming we are within the cutlocus.

To prove (3.6.1) holds in the sense of distributions on all of M, we argue asfollows. For any nonnegative φ ∈ C∞(M) with compact support,∫

Mφ(x)HK(dg(x))dVg(x) =

∫ ∞

0

∫Cg(r)

φ(

expp(θ, r))

HK(r)Jg(θ, r)dΘ(θ)dr.

Given a unit vector θ ∈ TpM, let rθ be the largest value of r such that s 7→ γθ(s) =expp(θ, s) minimizes up to s = r. By the Fubini theorem, we have∫

Mφ(x)HK(dg(x))dVg(x) =

∫Sm−1

∫ rθ

0φ(

expp(θ, r))

HK(r)Jg(θ, r)drdΘ(θ).

Now for 0 < r < rθ , by Lemma 3.5.9 and (3.6.2),

HK(r)Jg(θ, r) ≥ H(θ, r)Jg(θ, r) =∂

∂rJg(θ, r).

Hence∫M

φ(x)HK(dg(x))dVg(x) ≥∫

Sm−1g

∫ rθ

0φ(

expp(θ, r)) ∂

∂rJg(θ, r)drdΘ(θ)

= −∫

Sm−1g

∫ rθ

0

∂

∂r

(φ expp

)(θ, r)Jg(θ, r)drdΘ(θ)+

∫Sm−1

g

φ(

expp(θ, rθ))

Jg(θ, rθ)dΘ(θ)

≥ −∫

Sm−1g

∫ rθ

0

∂

∂r

(φ expp

)(θ, r)Jg(θ, r)drdΘ(θ).

By the Gauss lemma we arrive at∫M

φ(x)HK(dg(x))dVg(x) ≥ −∫M⟨∇φ,∇rg⟩g dVg =

∫M

rg∆φdVg,

where the last equality follows from the fact that rg is Lipschitz on M and thedivergence theorem holds for Lipschitz functions.


Using the inequality x coth x ≤ 1 + x yields

Corollary 3.6.2. If (M, g) is a complete Riemannian manifold with Ricg ≥ (n− 1)Kg,where K ≤ 0, then

(3.6.4) ∆dg ≤n− 1

dg+ (n− 1)

√|K|

in the sense of distributions. In particular, if (M, g) is a complete Riemannian manifoldwith Ricg ≥ 0, then for any p ∈ M

(3.6.5) ∆dg ≤m− 1

dg

in the sense of distributions.

Estimate (3.6.1) is sharp as can be seen from considering space forms of con-stant curvature −K. If K = 0, then (3.6.5) is sharp since on Euclidean space∆|x| = n−1

|x| .

3.6.2. Volume comparison theorem. A consequence of the Laplacian compar-ison theorem is

Theorem 3.6.3. (Bishop-Gromov volume comparison) If (M, g) is a complete Rie-mannian manifold with Ricg ≥ (n − 1)Kg, where K ∈ R, then for any p ∈ M, thevolume radio

Volg(Bg(p, r))VolK(BK(pK, r))

is a nonincreasing function of r, where pK is a point in the n-dimensional simply-connectedspace form of constant curvature K. In particular,

(3.6.6) Volg(Bg(p, r)) ≤ VolK(BK(pK, r))

for all r > 0. Given p ∈ M and r > 0, equality holds in (3.6.6) if and only if Bg(p, r) isisomorphic to BK(pK, r).

PROOF. Given a point pK ∈ MK, let ψpK : TpKMK \ 0 → Sn−1pK

be the stan-dard projection ψpk (V) := V/|V|pK . The volume element of the space form satis-fies

dVK :=√

det(gK)dθ1K ∧ · · · ∧ dθn−1

K ∧ drK = sn−1K (rg)dσK ∧ drK,

where dσK is the pull-back by ψpK exp−1pK

of the standard volume form on the unit

sphere Sn−1pK

. If (θi)1≤i≤n−1 are coordinates on Sn−1pK

, then

θiK = θi ψ exp−1

pK, i = 1, · · · , n− 1.

FromdσK = (ψ exp−1

pK)∗(

dθ1 ∧ · · · ∧ dθn−1)= dθ1

K ∧ · · · ∧ dθn−1K ,

we get

JK :=√

det(gK) = sn−1K (rg).


When K ≤ 0 the above formula holds for all rg > 0 and when K > 0 we need toassume rg ∈ (0, π/

√K).

Now we consider a Riemannian manifold (M, g) with Ricg ≥ (n − 1)Kg.From (3.5.19) and (3.5.40) we obtain

(3.6.7)∂

∂rgln

√det(g)

sn−1K (rg)

≤ 0.

Assume that the coordinates (θi)1≤i≤n−1 on Sn−1p are such that limr→0+

1r

∂∂θi :=

ei ∈ TpM are orthonormal. Then we have

limrg→0+

√det(g)

sn−1K (rg)

= 1

from which we conclude

(3.6.8) Jg ≤ sn−1K (rg).

Without making any normalizing assumption on the coordinates (θi)1≤i≤n−1, thissays

Jg(θg, rg)dΘg(θg) ≤ sn−1K (rg)dσSg(p,r)(θg).

Equivalently,J(θ, r)dΘ(θ) ≤ sm−1

K (r)dσSn−1p

(θ).

This is the infinitesimal are comparison formula which gives us

(3.6.9) dVg ≤ dVK.

Integrating this proves (3.6.6), at least within the cut locus. To see that this resultholds on the whole manifold, we argue as follows. Let

Cg(r) := V ∈ TpM : |V|g(p) = 1 and γV(s) = expp(sV), s ∈ [0, r], is minimizing.

Note that Cg(r2) ⊂ Cg(r1) for r1 ≤ r2. since the cut locus of p has measure zeroand exp∗p(dVg) = JdΘ ∧ dr inside the cut locus of p, for any integrable function φ

on a geodesic ball Bg(p, r) we have∫Bg(p,r)

φ(x)dVg(x) =∫ r

0

(∫Cg(r)

φ(

expp(θ, r))

J(θ, r)dΘ(θ)

)dr.

In particular,

Volg(Bg(p, r)) =∫

Bg(p,r)dVg =

∫ r

0

∫Cg(r)

exp∗p(dVg)

=∫ r

0

(∫Cg(r)

J(θ, r)dΘ(θ)

)dr ≤

∫ r

0

(∫Cg(r)

sn−1K (r)dσSn−1

p(θ)

)dr

≤∫ r

0

(∫Sn−1

p


p(θ)

)dr

=∫ r

0

(∫Sn−1

pK


pK(θ)

)dr = VolK(BK(pK, r)).

This completes the proof of (3.6.6).


Corollary 3.6.4. If (M, g) is a complete Riemannian manifold with Ricg ≥ 0, then forany p ∈ M, the volume ratio

Volg(Bg(p, r))rn

is a nonincreasing function of r. Since

limr→0

Volg(Bg(p, r))rn = ωn,

we have

(3.6.10)Volg(Bg(p, r))

rn ≤ ωn

for all r > 0, where ωn is the volume of the Euclidean unit n-ball.

Corollary 3.6.5. (Volume characterization of Rn) If (M, g) is a complete noncompactRiemannian manifold with Ricg ≥ 0 and if for some p ∈ M

limr→∞

Volg(Bg(p, r))rn = ωn,

then (M, g) is isomorphic to Euclidean space.

The Bishop-Gromov volume comparison theorem has been generalized to therelative volume comparison theorem. Let (M, g) be a complete Riemannian man-ifold and p ∈ M. Given a measurable subset Γ of the unit sphere Sn−1

p ⊂ TpMand 0 < r ≤ R < ∞, define the annular-type region:

(3.6.11) AΓg,r,R(p) :=

x ∈ M : r ≤ dg(p, x) ≤ R

and there exists a unit speedminimal geodesic γ from

γ(0) = p to xsatisfying γ′(0) ∈ Γ

⊂ Bg(p, R) \ Bg(p, r).

Note that id Γ = Sn−1p , then

ASn−1

pg,r,R(p) = Bg(p, R) \ Bg(p, r).

Given K ∈ R and a point pK in the n-dimensional simply-connected space form(MK, gK) of constant curvature K, let AΓ

gK ,r,R(pK) denote the corresponding set inthe space form.

Theorem 3.6.6. (Bishop-Gromov relative volume comparison theorem) Supposethat (M, g) is a complete Riemannian manifold with Ricg ≥ (n− 1)Kg. If 0 ≤ r ≤ R ≤S, r ≤ s ≤ S and if Γ ⊂ Sg−1

p is a measurable subset, then

(3.6.12)Volg

(AΓ

g,s,S(p))

VolK

(AΓ

gK ,s,S(pK)) ≤ Volg

(AΓ

g,r,R(p))

VolK

(AΓ

gK ,r,R(pK)) .


Taking r = s = 0 and Γ = Sn−1p yields Theorem 3.6.3.

Corollary 3.6.7. (Yau) Let (M, g) be a complete noncompact Riemannian manifoldwith nonnegative Ricci curvature. For any point p ∈ M, there exists a constantC = C(g, p, m) > 0 such that for any r ≥ 1

(3.6.13) Volg(Bg(p, r)) ≥ Cr.

PROOF. Let x ∈ M be a point with dg(p, x) = r ≥ 2. By Theorem 3.6.6 wehave

Volg

(ASn−1

xg,r−1,r+1(x)

)Volg

(ASn−1

xg,0,r−1(x)

) ≤ VolK

(ASn−1

xgK ,r−1,r+1(pK)

)VolK

(ASn−1

xgK ,0,r−1(pK)

)giving us

Volg(Bg(x, r + 1))−Volg(Bg(x, r− 1))Volg(Bg(x, r− 1))

≤ (r + 1)n − (r− 1)n

(r− 1)n ≤ C(n)r

for some constant C(n) depending only on n. Since Bg(p, 1) ⊂ Bg(x, r + 1) \Bg(x, r− 1) and Bg(x, r− 1) ⊂ Bg(p, 2r− 1), it follows that

Volg(Bg(p, 2r− 1)) ≥ Volg(Bg(x, r− 1)) ≥Volg(Bg(p, 1))

C(n)r.

We have proved the corollary for r ≥ 3. Clearly it is then true for any r ≥ 1.

If Ricg ≥ 0, then by (3.5.22) and (3.6.5) we have

(3.6.14) H = ∆dg ≤n− 1

dg.

Hence the area Ag(r) of the distance sphere Sg(p, r) satisfies

ddr

Ag(r) =∫

Sg(p,r)Hdσ ≤

∫Sg(p,r)

n− 1rg

dσ =n− 1

rAg(r).

Integrating this yields

Ag(s) ≤ Ag(r)sn−1

rn−1 , s ≥ r.

Therefore

(3.6.15) Volg(Bg(p, r)) =∫ r

0Ag(ρ)dρ ≥

∫ r

0Ag(r)

ρn−1

rn−1 dρ =rn

Ag(r),

we obtain

(3.6.16) Ag(s) ≤ nVolg(Bg(p, r))

rn sn−1, s ≥ r,

or

(3.6.17)Ag(s)

nωnsn−1 ≤Volg(Bg(p, r))

ωnrn , s ≥ r.


Let

(3.6.18) AVRg(p) := limr→∞

Volg(Bg(p, r))ωnrn

be the asymptotic volume ratio. The asymptotic volume ratio is an important in-variant of the geometry at infinity of a complete noncompact manifold with non-negative Ricci curvature.

Corollary 3.6.8. On a complete noncompact Riemannian manifold with nonnegativeRicci curvature we have

(3.6.19)Ag(s)

mωnsn−1 ≥ AVRg(p)

for any s ≥ 0.

PROOF. Since Ag(s)/sn−1 is nonincreasing, it follows that

Volg(Bg(p, s))−Volg(Bg(p, r))ωn(sn − rn)

=1

ωn(sn − rn)

∫ s

rAg(ρ)dρ

≤ 1ωn(rn − sn)

∫ s

rAg(r)

ρn−1

rn−1 dρ =Ag(r)

ωnrn−1

∫ sr ρn−1dρ

rn − sn =Ag(r)

nωnrn−1

for any s ≥ r. Letting s→ ∞ yields (3.6.19). 3.6.3. Hessian comparison theorem. The Hessian comparison theorem roughly

says that the larger the curvature, the smaller the Hessian of the distance function.

Proposition 3.6.9. (Hessian comparison theorem–general version) Let i = 1, 2.Let (Mi, gi) be complete Riemannian manifolds, let γi : [0, L] → Mi be geodesicsparametrized by arc length such that γi does not intersect the cur locus of γi(0), andlet dgi (·) := dgi (γi(0)). If for all t ∈ [0, L] we have

Secg1(V1 ∧ γ1(t)) ≥ Secg2(V2 ∧ γ2(t))

for all unit vectors Vi ∈ Tγi(t)Mi perpendicular to γi(t), then

(3.6.20) ∇2g1

dg1(X1, X1) ≤ ∇2g2

dg2(X2, X2)

for all Xi ∈ Tγi(t)Mi perpendicular to γi(t) and t ∈ (0, L].

We shall prove the following special case of the above result, namely compar-ing to constant curvature spaces.

Theorem 3.6.10. (Hessian comparison theorem–special case) Let (M, g) be a com-plete Riemannian manifold with Secg ≥ K. For any point p ∈ M the distance functionrg(x) := dg(p, x) satisfies

(3.6.21) ∇i∇jrg = hij ≤1

m− 1HK(rg)gij


at all points where rg is smooth (i.e., away from p and the cut locus). On all of M theabove inequality holds in the sense of support functions.

PROOF. From (3.5.27), we have

∇ ∂∂rg

h ≤ −Kg− h2

along a geodesic ray γ : [0, L)→M emanating from p. We claim that

(3.6.22) h(rg, θg) ≤1

m− 1HK(rg)g(rg, θg).

Indeed, given any unit vector V at p, we parallel translate it along γ. Let V(rg) :=V(γ(rg)); then |V(rg)|g(γ(rg)) = 1 and ∇∂/∂rg V(rg) = 0. Hence

ddrg

[h(V(rg), V(rg))

]= ∇ ∂

∂rgh(V(rg), V(rg)) + 2h

(∇ ∂

∂rgV(rg), V(rg)

)≤ −K|V(rg)|g(γ(rg)) − [h(V(rg), V(rg))]

2 = −K− [h(V(rg), V(rg))]2.

From (3.5.20) and (3.5.21) we have

h(V(rg), V(rg))−HK(rg)

n− 1=

1rg

+ O(rg)−1rg−O(rg) = O(rg).

Consequently

h(V(rg), V(rg))−HK(rg)

n− 1

≤[

h(V(ϵ), V(ϵ))− HK(ϵ)

n− 1

]exp

[−∫ r

ϵ

(HK(s)n− 1

+ h(V(s), V(s)))

ds]

which gives

h(V(rg), V(rg))−1

n− 1HK(rg) ≤ 0

for all rg > 0. Hence

∇i∇jrg = hij ≤1

n− 1HK(rg)gij

inside the cut locus when Secg ≥ K.

Note that the Hessian of the distance function is the second fundamental formof the distance sphere, which in turn is the radial derivative of the metric. yieldinginformation about the inner products of the Jacobi fields ∂

∂θig:

(3.6.23) ∇i∇jrg = hij =12

∂

∂rggij =

12

∂

∂rg

⟨∂

∂θig

,∂

∂θjg

⟩g

.

If J1 and J2 are Jacobi fields along a geodesic γ : [0, L] → M without conjugatepoints and if Ji(0) = 0 and ⟨(∇g)γ Ji(0), γ(0)⟩g(γ(0)) = 0 for i = 1, 2, then we have

(3.6.24)12

∂

∂rg⟨J1, J2⟩g = ∇J1∇J2 rg = h(J1, J2).


Corollary 3.6.11. Let (M, g) be Riemannian manifold with Secg ≥ K and let γ :[0, L] → M be a unit speed geodesic. If J is a Jacobi field along γ J(0) = 0 and⟨∇γ J(0), γ(0)⟩g(γ(0)) = 0, then

(3.6.25) |J(rg)|g(γ(rg)) ≤∣∣∣∇γ(0) J(0)

∣∣∣g(γ(0))

sK(rg).

PROOF. By our hypotheses, ⟨J(rg), γ(rg)⟩g(γ(rg)) = 0 for all rg ≥ 0. From(3.6.22) and (3.6.24),

∂

∂rg

(|J(rg)|g(γ(rg))

sK(rg)

)=

∂

∂rg

⟨J(rg), J(rg)⟩1/2g(γ(rg))

sK(rg)

=

12|J(rg)|g(γ(rg))

∂∂rg⟨J(rg), J(rg)⟩g(γ(rg))sK(rg)− |J(rg)|g(γ(rg))s

′K(rg)

s2K(rg)

=1

|J(rg)|g(γ(rg))sK(rg)h(J(rg), J(rg))−

s′K(rg)

sK(rg)

|J(rg)|g(γ(rg))

sK(rg)

=

[h

(J(rg)

|J(rg)|g(γ(rg)),

J(rg)

|J(rg)|g(γ(rg))

)−

HK(rg)

n− 1

]|J(rg)|g(γ(rg))

sK(rg)≤ 0.

The result follows from limrg→0 |J(rg)|g(γ(rg))/sK(rg) = |∇γ(0) J(0)|g(γ(0)).

Remark 3.6.12. Suppose that (M, g) is a Riemannian manifold with constant sec-tional curvature K. If J is a Jacobi field along a unit speed geodesic γ with J(0) = 0and ⟨∇γ J(0), γ(0)⟩g(γ(0)) = 0, then

|J(rg)|g(γ(rg)) =∣∣∣∇γ(0) J(0)

∣∣∣g(γ(0))

sK(rg).

In general we have the expansion

|J(rg)|2g(γ(rg))= r2

g −13

⟨Rmg

(∇γ(0) J(0), γ(0)

)γ(0),∇γ(0) J(0)

⟩g(γ(0))

r4g

+ O(r5g).(3.6.26)


Finally we consider the Hessian in spherical coordinates. We have

∇n∇n =∂2

∂r2g− Γa

nn∂

∂xa =∂2

∂r2g

,

∇n∇i =∂2

∂rg∂θig− Γa

ni∂

∂xa =∂2

∂rg∂θig− hj

i∂

∂θjg

,

∇i∇j =∂2

∂θig∂θ

jg− Γa

ij∂

∂xa = ∇Si ∇

Sj + hij

∂

∂rg,

where ∇S is the intrinsic covariant derivative of the hypersurface Sg(p, r). In par-ticular, if f = f (rg) is a radial function, then

∇n∇n f =∂2 f∂r2

g, ∇n∇i f = 0, ∇i∇j f = hij

∂ f∂r

.

3.6.4. Mean value inequalities. The following mean value inequality that is aconsequence of the Laplacian comparison theorem, has an application in the proofof the splitting theorem.

Proposition 3.6.13. (Mean value inequality for Ricg ≥ 0) If (M, g) is a completeRiemannian manifold with Ricg ≥ 0 and if f ≤ 0 is a Lipschitz function with ∆ f ≥ 0 inthe sense of distribution, then for any x ∈ M and 0 < r < injg(x),

(3.6.27) f (x) ≤ 1ωmrm

∫Bg(x,r)

f dVg.

PROOF. By the divergence theorem for Lipschitz functions, we have

0 ≤ 1rn−1

∫Bg(x,r)

∆ f dVg =∫

∂Bg(x,r)

∂ f∂r

1rn−1

√det(g)dΘg,

where dΘg = dθ1g ∧ · · · ∧ dθn−1

g . Since

∂

∂rg

√det(g)rn−1

g=

∂

∂rg

(Jg

rn−1g

)=

∂∂rg

Jg · rn−1g − Jg(n− 1)rn−2

g

r2n−2g

=

∂∂rg

Jg · rg − Jg(n− 1)

rng

≤n−1rg

Jgrg − Jg(n− 1)

rng

= 0

from ∆rg = H = ∂∂rg

ln Jg ≤ n−1rg

and f ≤ 0, we have

0 ≤∫

∂Bg(x,r)

(∂ f∂r

√det(g)rn−1 + f

∂

∂r

√det(g)rn−1

)dΘg

=∫

∂Bg(x,r)

∂

∂r

(f√

det(g)rn−1

)dΘg =

ddr

(1

rn−1

∫∂Bg(x,r)

f dσg

),


where dσg =√

det(gij)dΘg. Since

limr→0

1rn−1

∫∂Bg(x,r)

f dσg = nωn f (x),

where mωn is the volume of the unit (n− 1)-sphere, integrating the above inequal-ity over [0, s] yields

nωn f (x) ≤ 1sn−1

∫∂Bg(x,s)

f dσg.

Integrating this again over [0, r] implies

f (x)rn

n≤ 1

nωn

∫Bg(x,r)

f dVg

which is the desired inequality (3.6.27).

Proposition 3.6.14. (Mean value inequality for Secg ≤ H) Suppose that (M, g) isa complete Riemannian manifold with Secg ≤ H in a ball Bg(x, r) where r < injg(M).If f ∈ C∞(M) is subharmonic, i.e., if ∆ f ≥ 0, and if f ≥ 0 onM, then

(3.6.28) f (x) ≤ 1VH(r)

∫Bg(x,r)

f dVg,

where VH(r) is the volume of a ball of radius r in the complete simply-connected manifoldof constant sectional curvature H.

3.6.5. Rauch comparison theorem. More generally, applying standard ODEcomparison theory to the Jacobi equation, one has the following

Theorem 3.6.15. (Rauch comparison theorem) Let (M, g) and (M, g) be Riemann-ian manifolds with the same dimension n and let γ : [0, L]→M and γ : [0, L]→ M beunit speed geodesics. Suppose that γ has no conjugate points and for any r ∈ [0, L] andany X ∈ Tγ(r)M, X ∈ Tγ(r)M, we have

Secg (X ∧ γ(r)) ≤ Secg (X ∧ ˙γ(r)) .

If J and J are Jacobi fields along γ and γ with J(0) and J(0) tangent to γ and γ, and if

|J(0)|g(γ(0)) = | J(0)|g0(γ(0)),⟨∇γ(0) J(0), γ(0)

⟩g(γ(0))

=⟨∇ ˙γ(0) J(0), ˙γ(0)

⟩g(γ(0))

,∣∣∣∇γ(0) J(0)∣∣∣g(γ(0))

=∣∣∣∇ ˙γ(0) J(0)

∣∣∣g(γ(0))

,

then

(3.6.29) |J(r)|g(γ(r)) ≥ | J(r)|g(γ(r)).

3.7. MANIFOLDS WITH NONNEGATIVE CURVATURE 195

Corollary 3.6.16. (Cartan-Hadamard theorem) If (M, g) is a complete Riemanniann-manifold with nonpositive sectional curvature, then for any p ∈ M, the exponentialmap expp : TpM → M is a covering map. In particular, the universal cover of M isdiffeomorphic to Euclidean space Rn.

3.7. Manifolds with nonnegative curvature

The structure of Riemannian manifolds with nonnegative curvature is verycomplicated.

3.7.1. The topological sphere theorem. Given a Riemannian manifold (M, g),let Secg(Π) denote the sectional curvature of a 2-plane Π ⊂ TpM where p ∈ M.The Rauch-Klingenberg-Berger topological sphere theorem says the following.

Theorem 3.7.1. (Topological sphere theorem) If (M, g) is a complete, simply-connected Riemannian n-manifold with 1

4 < Secg(Π) ≤ 1 for all 2-planes Π, thenM ishomeomorphic to the n-sphere. In particular, if m = 3, thenM3 is diffeomorphic to the3-sphere.

Recently, Brendle and Schoen showed that

Theorem 3.7.2. (Diffeomorphic sphere theorem) If (M, g) is a complete, simply-connected Riemannian n-manifold with 1

4 < Secg(Π) ≤ 1 for all 2-planes Π, thenM isdiffeomorphic to the n-sphere.

There is not much known about general closed Riemannian manifolds withpositive sectional curvature.

Problem 3.7.3. (Hopf conjecture I) Does there exist a Riemannian metric on S2 × S2

with positive sectional curvature?

Problem 3.7.4. (Hopf conjecture II) Prove that if (M2m, g) is a closed, even-dimensional Riemannian manifold with positive sectional curvature, then χ(M2m) > 0.

Note that any closed, odd-dimensional manifold has χ(M2m+1) = 0. The caseof complete noncompact manifolds with positive sectional curvature is simpler.

3.7.2. Cheeger-Gromoll splitting theorem and soul theorem. In the studyof manifolds with nonnegative curvature, often (especially when the curvatureis not strictly positive) the manifolds split as the product of a lower-dimensionalmanifold with a line.


A geodesic line is a unit speed geodesic γ : (−∞, ∞) → M such that thedistance between any points on γ is the length of the arc of γ between those twopoints; that is, for any s1, s2 ∈ (−∞, ∞), dg(γ(s1), γ(s2)) = |s2 − s1|. A unit speedgeodesic β : [0, ∞) → M is a geodesic ray if it satisfies the same condition asabove. Given a geodesic ray β : [0, ∞)→M, the Busemann function

(3.7.1) bβ :M−→ R

associated to β is defined by

(3.7.2) bβ(x) := lims→∞

(s− dg(β(s), x)

).

Remark 3.7.5. (1) In Euclidean space the Busemann function is linear. For anyunit vector V ∈ Rn, the Busemann function bγV associated to the geodesic rayγV : [0, ∞)→ Rn defined by γV(s) := sV is the linear function given by

bγV (x) = ⟨x, V⟩for all x ∈ Rn.

(2) The Busemann function is well-defined, finite, and Lipschitz.(3) |∇bβ|g = 1 at points where it is C1.(4) If β is a geodesic ray in a Riemannian manifold with Ricg ≥ 0, then ∆bβ ≥ 0

in the sense of distributions. Indeed, using (3.6.5) yields

∆bβ(x) ≥ − m− 1lims→∞ dg(β(s), x)

= 0.

Theorem 3.7.6. (Cheeger-Gromoll) Suppose (M, g) is a complete noncompact Rie-mannian n-manifold with Ricg ≥ 0 and suppose that there is a geodesic line inM. Then(M, g) is isomorphic to R× (N , h) with the product metric, where (N , h) is a Riemann-ian (n− 1)-manifold with Rich ≥ 0.

PROOF. Given a geodesic line γ, consider the two Busemann functions bγ±associated to the geodesic rays γ± : [0, ∞) → M defined by γ±(s) = γ(±s) fors ≥ 0. Since Ricg ≥ 0, we have ∆bγ± ≥ 0 in the sense of distributions by Remark3.7.5 and hence ∆(bγ+ + bγ−) ≥ 0. From dg(γ(s), γ(−s)) = 2s, we note that forany x ∈ M

bγ+(x) + bγ−(x) = lims→∞

[2s− dg(γ(s), x)− dg(γ(−s), x)

]≤ lim

s→∞

[2s− dg(γ(s), γ(−s))

]= 0.

Using Proposition 3.6.14, we obtain

0 = bγ+(x) + bγ−(x) ≤ 1ωnrn

∫Bg(x,r)

[bγ+ + bγ− ]dVg ≤ 0

for any x ∈ γ and 0 < r < injg(x). Hence bγ+ + bγ− ≡ 0 in a neighborhood of γ.By applying the mean value inequality again, we see that the set of points in

M where bγ+ + bγ− = 0 is open. Since this set is also closed and nonempty, we


have bγ+ + bγ− ≡ 0 onM and hence also ∆(bγ+ + bγ−) ≡ 0. Since ∆bγ± ≥ 0, thisimplies ∆bγ± = 0 in the sense of distributions. Standard regularity theory of PDEnow implies bγ± is smooth. Therefore, |∇bγ± |g ≡ 1 by Remark 3.7.5. Since ∇bγ±is a nonzero parallel gradient vector field on M, (M, g) splits as a Riemannianproduct R× (N , h) where N = x ∈ M : bγ+(x) = 0.

A submanifold S ⊂ M is totally convex if for every x, y ∈ S and any geodesicγ (not necessarily minimal) joining x to y we have γ ⊂ S . We say that S is totallygeodesic if its second fundamental form is zero. In particular, a path in a totallygeodesic submanifold S is a geodesic in S if and only if it is a geodesic inM.

Given a noncompact manifold (M, g), we say that a submanifold is a soul ifit is a closed, totally convex, totally geodesic submanifold such thatM is diffeo-morphic to its normal bundle.

Theorem 3.7.7. (Cheeger-Gromoll, 1972) Let (M, g) be a complete noncompact Rie-mannian n-manifold with nonnegative sectional curvature. Then there exists a soul. If thesectional curvature is positive, then the soul is a point (e.g.,M is diffeomorphic to Rn).

Furthermore Sharafurdinov proved that any two souls are isometric.

Theorem 3.7.8. (Soul conjecture)(Perelman, 1994) If (M, g) is a complete noncom-pact Riemannian manifold with nonnegative sectional curvature everywhere and positivesectional curvature at some point, then the soul is a point.

Another fundamental result about noncompact manifolds with positive sec-tional curvature is the following.

Theorem 3.7.9. (Toponogov, 1959) If (M, g) is a complete noncompact Riemannianmanifold with positive sectional curvature bounded above by K, then

(3.7.3) inj(M, g) ≥ π√K

.

Moreover,M is diffeomorphic to Euclidean space.

3.7.3. Toponogov comparison theorem. As a consequence of Section 3.3 wehave the following

Lemma 3.7.10. Let (M, g) be a complete Riemannian manifold with nonnegative sec-tional curvature and p ∈ M. If β : (a, b) → M is a unit speed geodesic, then thefunction ϕ : (a, b)→ R defined by

ϕ(r) := r2 − d2g(p, β(r))

is convex.


PROOF. Given r0 ∈ (a, b), let γr : [0, L]→M be a 1-parameter family of pathsfrom p to β(r) with γr0 : [0, L]→M a unit speed minimal geodesic from p to β(r0)and

∂

∂r

∣∣∣r=r0

γr(s) =sL

V(γr0(s)),

where V is the parallel translation of β(r0) ∈ Tγr0 (L)M along γ. Since |V|2g = 1, itfollows from the proof of Lemma 3.3.12 that (since Secg ≥ 0)

d2

dr2

∣∣∣r=r0

(r2 − L2

g(γr))≥ 2− 2 = 0.

Since

r2 − d2g(p, β(r)) ≥ r2 − L2

g(γr), r20 − d2

g(p, β(r0)) = r20 − L2

g(γr0),

we conclude that ϕ is convex.

In general, if ϕ : (a, b) → R is a Lipschitz function such that for all r0 ∈ (a, b)there exists a C2-function ψr0(r) defined in a neighborhood of r0 with ψr0(r) ≤ϕ(r), ψr0(r0) = ϕ(r0) and d2

dr2 |r=r0 ψr0(r) ≥ 0, then ϕ is convex.Using Lemma 3.7.10, we can give a proof of the Toponogov comparison the-

orem.

Theorem 3.7.11. (Toponogov comparison theorem–Secg ≥ 0)(Toponogov, 1959)Let (M, g) be a complete Riemannian manifold with nonnegative sectional curvature andlet α : [0, A] → M be a unit speed minimal geodesic joining p to q. If β : [0, B] → Mis a unit speed geodesic with β(0) = q and if θ ∈ [0, π] is the angle between β(0) and−α(A), then

d2g(p, β(r)) ≤ r2 + A2 − 2rA · cos(θ)

for all r ∈ [0, B]. In particular,

(3.7.4) d2g(p, β(B)) ≤ A2 + B2 − 2AB · cos(θ).

By the law of cosines, equality is attained for Euclidean space. That is, the right-hand sideof (3.7.4) is the length squared of the side in the corresponding Euclidean triangle with thesame A, B and θ.

PROOF. For ϵ > 0, let

fϵ(r) := r2 − d2g(p, β(r)) + A2 − 2Ar · cos(θ) + ϵr.

By Lemma 3.7.10, fϵ is convex. We also have

fϵ(0) = −d2g(p, q) + A2 = −L2

g(α) + A2 = −A2 + A2 = 0

because α is a unit speed minimal geodesic. By a first variation argument (we mayassume dg(p, ·) is smooth at q. Otherwise, we can apply Calabi’s trick)

∂

∂r

∣∣∣r=0

fϵ(r) =(

2r− 2dg(p, β(r))⟨∇dg(p, β(r)), β(r)

⟩g − 2A · cos(θ) + ϵ

)r=0

= ϵ− 2A⟨α(A), β(0)⟩g − 2A · cos(θ) = ϵ > 0,


fϵ(r) > 0 for r > 0 small enough, depending on ϵ. Since fϵ is convex, we concludethat fϵ(r) > 0 for all r ∈ (0, B]. In particular, limϵ→0 fϵ(r) ≥ 0 for all r ∈ (0, B],which proves the theorem.

More generally, we have the following statements of the Toponogov compari-son theorems for manifolds with a sectional curvature lower bound.

A geodesic triangle is a triangle (i.e., three vertices joined by three paths)whose sides are geodesics. The triangle version says that a triangle in a mani-fold has larger angles than the corresponding triangle with the same side lengthsin the simply-connected constant curvature space. Given a triangle (p, q, r), ]pqrdenote the angle at q.

Theorem 3.7.12. (Toponogov comparison theorem–triangle version) Let (M, g)be a complete Riemannian manifold with Secg ≥ K. Let be a geodesic triangle withvertices (p, q, r), sides qr, rp, pq, corresponding lengths a = Lg(qr), b = Lg(rp), c =Lg(pq) such that a ≤ b + c, b ≤ a + c, c ≤ a + b (for example, when all of the geodesicsides are minimal), and interior angles α = ]rpq, β = ]pqr, γ = ]qrp, where α, β, γ ∈[0, π]. If the geodesics qr and rp are minimal, and c ≤ π/

√K in the case where K > 0

(no assumption on c when K ≤ 0), then there exists a geodesic triangle = ( p, q, r) inthe complete, simply-connected 2-manifold of constant Gauss curvature K with the sameside lengths (a, b, c) and such that we have the following comparison of the interior angles:

α ≥ α := ]r pq, β ≥ β := ]pqr.

A geodesic hinge consists of a pair of geodesic segments emanating from apoint, called the vertex, making an angle at the vertex. The hinge version saysthat a hinge in a manifold has a smaller distance between its endpoints than thecorresponding hinge in the constant curvature space with the same “side-angle-side”.

Theorem 3.7.13. (Toponogov comparison theorem–hinge version) Suppose(M, g) is a complete Riemannian manifold with Secg ≥ K. Let ∠ be a geodesic hingewith vertices (p, q, r), sides qr, rp, and interior angle ]qrp ∈ [0, π] inM. Suppose thatqr is minimal and that Lg(rp) ≤ π/

√K if K > 0. Let ∠′ be a geodesic hinge with ver-

tices (p′, q′, r′) in the simply-connected space of constant curvature K with the same sidelengths Lg(q′r′) = Lg(qr), Lg(r′p′) = Lg(rp) and same angle ]q′r′p′ = ]qrp. Thenwe have the following comparison of the distance between the endpoints of the hinges:

dg(p, q) ≤ dK(p′, q′)

where dK denotes the distance in the simply-connected space of constant sectional curva-ture K.


3.8. Space of metric measure spaces

3.9. Ricci flow


CHAPTER 4

Kahler manifolds

4.1. Complex manifolds

4.2. Kahler manifolds

4.3. Calabi-Yau manifolds

4.4. Compact complex surfaces

4.5. Kahler-Ricci flow


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math.seu.edu.cn · abstract. main references: [1] choquer-bruhat, yvonne; dewitt-morette, c.;...

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