The General Theorem of Stokes andApplications

Orvar Lorimer OlssonBachelor’s thesis2019:K10

Faculty of ScienceCentre for Mathematical SciencesMathematics

CE

NT

RU

MSC

IEN

TIA

RU

MM

AT

HE

MA

TIC

AR

UM

Abstract

In this Bachelor’s thesis we give an account for the theory needed to state andprove The General Theorem of Stokes.

Theorem. For a compact oriented m-dimensional manifold M with boundary ∂Mand a differentiable (m− 1)-form ω on M, we have the following integral identity∫

M

dω =

∫∂M

ω,

where the boundary ∂M has the induced orientation.

Further we show some applications of the above result with specific focus onhypersurfaces, for which we show the following result.

Theorem. Let M be a compact hypersurface without boundary immersed in theEuclidean Rm+1 with shape operator S = dN . Then we have the following identities∫

M

Hk(S)dVolM =

∫M

Hk+1(S)〈N, p〉dVolM , for k = 0, . . . , (m− 1),

where Hk(S) denotes the k-sectional mean curvature, that is

Hk(S)p =

(m

k

)−1∑1≤i1<···<ik≤m

λi1 · · ·λik ,

where λj are the eigenvalues of the shape operator Sp.

Throughout this work it has been my firm intention to give reference to the statedresults and credit to the work of others. All theorems, propositions, lemmas andexamples left unmarked are assumed to be too well known for a reference to be given.

Acknowledgements

I wish to thank my family for their support and encouragement throughout mystudy. Further I wish to acknowledge the help provided by the staff at The Libraryof Mathematics at Lund University. They provided me with good support, andhelped me find my way to useful textbooks and reference material. Finally I thankmy supervisor Sigmundur Gudmundsson for his support and patience throughoutthis project.

Orvar Lorimer Olsson

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

1 Differential Forms 11.1 The Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Dual Space, Tensors and k-Vectors . . . . . . . . . . . . . . . . . 41.3 The Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 The Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Integration of Forms, Singular Cubes and Chains 112.1 Integration on Singular Cubes and Chains . . . . . . . . . . . . . . . 112.2 The Boundary Operator . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The General Theorem of Stokes on Singular Chains . . . . . . . . . . 14

3 Manifolds 173.1 Definitions and Well Known Results . . . . . . . . . . . . . . . . . . . 173.2 Manifolds and Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Integration on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 The General Theorem of Stokes on Manifolds With Boundary . . . . 21

4 Applications 254.1 Classical Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 Isometric Ovaloids . . . . . . . . . . . . . . . . . . . . . . . . 314.2.2 The k-Sectional Curvatures of Hypersurfaces . . . . . . . . . . 36

0.1 Introduction

The general theorem of Stokes on manifolds with boundary states the deceivinglysimple formula ∫

M

dω =

∫∂M

ω,

where ω is a differentiable (m− 1)-form on a compact oriented m-dimensional man-ifold M.

To fully understand the formula though, we need to describe all the notions itcontains.

In Chapter 1, we define k-vectors, the differentiable forms and the exterior deriva-tive d that acts upon them.

In Chapter 2, singular cubes and chains are defined in order to define integrationof forms on cubes and chains. Further the boundary operator ∂ is defined for chains.The chapter is completed by doing the main work in proving Stokes’ general theoremby proving it for singular chains.

Chapter 3, defines integration of forms ω over a manifold M with boundary ∂Musing the singular cubes together with a partition of unity. With all this well definedwe are able to properly state and prove the general theorem of Stokes on manifoldswith boundary.

Our account of this theory is heavily based on the books [1] of Spivak, [2] ofFlanders, and [3] of doCarmo. Most of the definitions, theorems and proofs will befound within these publications.

The last Chapter is devoted to applications of the theorem, first as a generalisa-tion of classical results of vector calculus, and secondly to derive results regardingovaloids and hypersurfaces. The section on isometric ovaloids can be seen as a write-up of Gudmundsson’s lecture notes [8] on the subject. The result on curvature ten-sors on hypersurfaces is a closely related generalisation. To easier understand thislast section reader should be familiar to notation and central notions in RiemannianGeometry as presented in [7] of Gudmundsson.

We do not claim any significant contribution to the theory. This text is rathera way of showing my interpretation as a merged result of taking part of the afore-mentioned works.

0

Chapter 1

Differential Forms

Vectors and vector fields are important mathematical objects, especially in differen-tial geometry. In this chapter, we seek to generalise these concepts further. Froma physical point of view, a vector can be seen as a direction and speed or a signedsize. These line elements or singular directions form a vector space at each point.Similarly, if we instead of choosing one direction at a time, chose two and consid-ering the ”speed” to be the rate of the change in area. We would form a space ofsurface elements spanning all possible oriented surface through that point, togetherwith a notion of ”speed” or ”signed size”. These will also form a vector space inthe mathematical sense, but a different one. We will call this the space of 2-vectors.Similarly, we can think of higher degree p-vectors in an n-dimensional space as anoriented p-dimensional object at that point, together with a ”signed size”. At thetop end of these objects, we find the n-dimensional object through an n-dimensionalspace i.e. a ”signed size measure” of the set of vectors in n-dimensions, which werecognise as the well known determinant function.

Just as we can study vector fields in standard vector calculus, we can also lendour attention to fields of these p-vectors in the n-dimensional space, which we callp-forms. But we need to construct a bit of theory, to get there. This chapter startswith constructing a language for the algebra we wish to use, and then connectingit with the objects that we are looking for. Throughout this work we will constructmultiple vector fields. These will be understood to be fields over R with the standardEuclidean metric. Many results are applicable for other fields, but the substitutionis then not to difficult for the reader.

1.1 The Exterior Algebra

In this section we closely follow the presentation of Flanders, see [2].Let V be a real vector space. Our goal is now for each k ∈ N to define a vectorspace

∧kV

called the space of k-vectors on V and k may be referred to as the degree of theelements. To start we set

∧0V = R, ∧1

V = V.

1

At this stage, our new notation might seem redundant, but these initial spaceswill later make more sense, when we have described what we actually mean by thesymbol∧.

Definition 1.1 (The exterior product). [2] Let · ∧ · be an operation such that foreach x, y, z ∈ V and a, b, c ∈ R we have

(i) (ax+ by) ∧ z = a(x ∧ z) + b(y ∧ z),

(ii) x ∧ (by + cz) = b(x ∧ y) + c(x ∧ z),

(iii) x ∧ x = 0,

(iv) x ∧ (y ∧ z) = (x ∧ y) ∧ z.

Then ∧ is called the wedge product, or the exterior product, and x ∧ y is called theexterior product of the vectors x and y.

The set of exterior products V ∧ V = x ∧ y | x, y ∈ V generates our nextlinear space. To get the full space, we simply need to add all possible finite sums ofthese elements and the algebraic closure will be assured. We denote

∧2V = spanx ∧ y | x, y ∈ V .

Since the wedge product is assumed to be associative we can comfortably make thefollowing formal definition for any k ≥ 2.

Definition 1.2. [2] For a positive integer k ≥ 2 we define the space

∧kV := spanx1 ∧ · · · ∧ xk : xi ∈ V .

With these spaces now defined, we see that there is no problem letting our oper-ation act on any k-vector.

Definition 1.3. [2] The exterior multiplication ∧ : ∧kV×∧l

V → ∧k+lV is

defined by

(x1 ∧ · · · ∧ xk) ∧ (y1 ∧ · · · ∧ yl) = x1 ∧ · · · ∧ xk ∧ y1 ∧ · · · ∧ yl.

From (i) − (iii) of Definition 1.1 we can directly derive some more interestingproperties.

Proposition 1.4. [2] For the operation ∧ we have the following properties, that forx, y ∈ V

(1) 0 ∧ x = 0,

(2) x ∧ y = −y ∧ x.

Proof. [2] The statement (1) is a direct consequence of the linearity. For (2) letx, y ∈ V , then

0 = (x+ y) ∧ (x+ y) = x ∧ y + x ∧ x+ y ∧ y + y ∧ x = x ∧ y + y ∧ x.

2

We can now summarise the properties of ∧ to be(a) distributive,(b) associative,(c) x1 ∧ · · · ∧ xp = 0 if the xi are linearly dependent,(d) x1 ∧ · · · ∧ xp change sign if any two xi are interchanged,

(e) λ ∧ µ = (−1)klµ ∧ λ for all λ ∈∧kV , µ ∈∧l

V .

The property (e) simply states that if two vectors are both of odd degree, theyanti-commute. Otherwise they commute. This is a bit vague at this point, but itwill be cleared up as we show how to construct a basis of these spaces, and calculatetheir dimensions.

We will now give a more detailed description of these spaces. For this we startby letting V be a real linear space of dimension n, so that we can start constructing

basis elements for∧kV . If k > n then any set of k vectors are linearly dependent,

so ∧kV = 0. Therefore these spaces will be of no interest, so we will assume that

k ≤ n. Let σ1 . . . σn be a basis for V so that two general element of x, y ∈ V canbe written as

x =∑

aiσi, y =∑

biσi ai, bi ∈ R.

We now get

x ∧ y =(∑

aiσi)∧(∑

biσi)

=∑

aibj(σi ∧ σj).

By using properties (c) and (d) of the exterior product we then obtain

x ∧ y =∑i<j

(aibj − ajbi)σi ∧ σj

and see that the 2-vectors σi ∧ σj, where 1 ≤ i < j ≤ n form a basis of∧2V . The

total number of basis elements will therefore be(n2

). In general, for any p, the set

of(nk

)k-vectors

σi1 ∧ · · · ∧ σik , 1 ≤ i1 < · · · < ip ≤ n

form a basis for∧kV , so a general element can be written as∑

1≤i1<···<ik≤n

ai1,...,ikσi1 ∧ · · · ∧ σik .[2]

Noticeable is that when k = n, there is only the one basis element σ1 ∧ · · · ∧ σn.

Consequently the space∧nV is one dimensional and any n-vector v1 ∧ · · · ∧ vn can

be written as a scalar multiple a of σ1 ∧ · · · ∧ σn, that is

v1 ∧ · · · ∧ vn = aσ1 ∧ · · · ∧ σn.

For a given basis σ we can thus form a mapping aσ : V n → R by this identification.Then aσ is multilinear and alternating, and aσ(σ) = 1. Using this insight andnoting how a function like aσ changes with linear transformations, we can constructan exterior algebra definition of the determinant function.

3

Definition 1.5. [2] For an n-dimensional vector space V , let A : V → V be a lineartransformation. Then we define a function gA of n variables on V by

gA : V n → ∧nV,

gA(v1, . . . , vn) = Av1 ∧ · · · ∧ Avn.

Since gA is multilinear and alternating, there is a linear transformation fA

fA : ∧nV → ∧n

V

such that fA(v1 ∧ · · · ∧ vn) = gA(v1, . . . , vn).

Since the space∧nV is only one dimensional any linear transformation is just a mul-

tiplication by a scalar. This interpretation serves as a definition for the determinantof A so that

Av1 ∧ · · · ∧ Avn = det(A)(v1 ∧ · · · ∧ vn).

This definition of the determinant of a linear operator is equivalent to the standarddefinition for matrices.

1.2 The Dual Space, Tensors and k-Vectors

On an n-dimensional vector space V the determinant can be considered as a functionV n → R that is multilinear and vanishes when the vectors are linearly dependent.We will start with a more general object known as a tensor, and then produce func-tions V k → R with k ≤ n that are similar to the determinant. It will then be clear,that the space of these functions, is an example of an exterior algebra as definedpreviously.

We start off by defining the dualspace and the notion of a dual basis.

Definition 1.6. [3] Let V be a vector space, then the dual space of V denoted byV ∗ is the set of all linear maps ϕ : V → R . If V has the basis v1, . . . , vn, then thebasis of the dual space, or the dual basis of v1, . . . , vn is the set of linear maps ϕisuch that

ϕi(vj) =

0, if i 6= j

1, if i = j.

This is a basis for all linear functions T : V → R also known as the space of 1-tensorson V and denoted J1(V ).

Definition 1.7. [1] For a vector space V , a k-tensor is a multilinear function T :V k → R. Further, for two k-tensors T, S, v ∈ V k and a ∈ R we define

(T + S)(v) = T (v) + S(v),

(aT )(v) = a · T (v).

Then the set of k-tensors with these distributive laws is a vector space called thespace of k-tensors denoted Jk(V ).

Associated to this is also an operation connecting the various tensor spaces.

4

Definition 1.8. [1] The tensor product ⊗ : Jk(V )×Jl(V )→ Jk+l(V ) is defined sothat for each T ∈ Jk(V ), S ∈ Jl(V ) and vi ∈ V we have

T ⊗ S(v1, . . . , vk, vk+1, . . . , vk+l) = T (v1, . . . , vk) · S(vk+1, . . . , vk+l).

It is easy to see that the tensor product satisfies the following distributive andassociative laws [1]

(T1 + T2)⊗ (S) = T1 ⊗ S + T2 ⊗ S,T ⊗ (S1 + S2) = T ⊗ S1 + T ⊗ S2,

(aT )⊗ S = T ⊗ (aS) = a(T ⊗ S),

(T ⊗ S)⊗R = T ⊗ (S ⊗R).

We also see that if T1, . . . , Tk ∈ J1(V ) then T1 ⊗ · · · ⊗ Tk ∈ Jk(V ). We will also seethat any k-tensor T ∈ Jk(V ) can be expressed in such a way. This makes it possibleto express a basis of Jk(V ) in terms of J1(V ) = V ∗.

Theorem 1.9. [1] Let v1, . . . , vn be a basis for V and ϕ1, . . . , ϕn its dual basis.Then the set of k-fold tensors

ϕi1 ⊗ · · · ⊗ ϕik : 1 ≤ i1 ≤ · · · ≤ ik ≤ n

forms a basis for Jk(V ), which therefore has the dimension nk.

This means that Jk(V ) =⊗k(V ∗). We would like to compare this space to

∧k(V ∗). Both operators ⊗ and ∧ are distributive and associative, but for the ∧

operator nullify the contribution of linearly dependent sets of vectors i.e. ϕ∧ϕ = 0.Define

Ldk(V ) = spanη1 ⊗ · · · ⊗ ηk : ηi ∈ V ∗are linearly dependent,

then Ldk(V ) is a subspace of Jk(V ). Further, for all T ∈ Jk(V ) we have

T ⊗ Ldj(V ) ⊂ Ldk+j(V ).

This means that for the union vector field J(V ) =⋃i J

i(V ) the set Ld(V ) =⋃i Ld

i(V ) forms the vector field equivalent to an ideal of an algebraic ring. And we

find that the quotient space is equivalent to∧(V ∗) =⋃i∧i

(V ∗).

Theorem 1.10. [1] Let ∼ be a relation between k-tensors on V defined by

T ∼ S if and only if S − T ∈ Ldk(V )

Then ∼ is an equivalence relation on the spaces of tensors also well defined for⊗. Further J(V )/Ld(V ) ∼= ∧(V ∗), where the operator ⊗ corresponds to ∧, also

Jk(V )/Ldk(V ) ∼=∧k(V ∗) for all non negative integers k.

1.3 The Differential Forms

Let us now place ourselves in Rn with the standard Euclidean metric. Let the mapsxi : Rn → R be the natural projection onto the i:th coordinate

xi(a1, . . . , an) = ai.

5

For each point p ∈ Rn, we have the tangent space TpRn isomorphic to Rn. Each ofthese are equipped with the canonical basis (e1)p, . . . , (en)p. Then the differentialsof the projection maps

(dxi)p : TpRn → Txi(p)R ∼= Rare linear maps given by the matrices

(dxi)p =( ∂xi∂x1

, . . . ,∂xi∂xn

)p.

Since∂xi∂xj

=

0, if i 6= j

1, if i = j

we have that

(dxi)p(ej) =

0, if i 6= j

1, if i = j. [3]

Thus by Definition 1.6 (dx1)p, . . . , (dxn)p is a basis for the dual space (TpRn)∗ andwe have for each v = (v1, . . . , vn) ∈ TpRn that (dxi)p(v) = vi. Just as we can form avector field, by to each point p in the space assign a vector v ∈ TpRn, we can nowform an equivalent structure in the dual space, these are the 1-forms.

Definition 1.11. [2] Let p be a point in Rn. Then an element λp ∈ (TpRn)∗ is calledan exterior 1-form at p and can be written as

λp =n∑i=1

ai(dxi)p, ai ∈ R.

Let U be an open subset of Rn. A 1-form in U is a map ω that associates to eachpoint p ∈ U a 1-form at that point ωp ∈ (TpRn)∗ and can be written, with respectto the basis, as

ωp =n∑i=1

αi(p)(dxi)p,

where αi : U → R.

If we now extend the definition of the exterior product ∧ to forms in the openset U , so that if ω, η are 1-forms, then

(ω ∧ η)p = ωp ∧ ηp for all p ∈ U. [1]

This makes (ω∧ η)p an element of∧2(TpRn)∗ and ω∧ η a map, that assign such an

element, to each point p in U . We will call ω ∧ η a 2-form in U.

Definition 1.12. [1, 2, 3] Let p be a point in Rn. Then an element λp ∈∧k(TpRn)∗

is called an (exterior) k-form at p. For an open subset U of Rn, a k-form in U is a

map ω that associates to each p ∈ U a k-form ωp ∈∧k(TpRn)∗, at that point, and

can be written with respect to the basis (dx1)p, . . . , (dxn)p as

ωp =∑

1≤j1<···<jk≤n

αj1,...,jk(p)(dxj1)p ∧ · · · ∧ (dxjk)p,

6

where the αj1,...,jk : Rn → R are some differentiable functions. This also means thata 0-form is a function U → R.

Further, for a point p ∈ Rn, given k vectors (v1, . . . , vk)p ∈ TpRn each expressedin a basis e1, . . . en as vi = (vi1, . . . , v

in), and a k-form ωp expressed in the dual

basis (dx1)p, . . . , (dxn)p as above, ωp(v1, . . . , vk) may be evaluated in harmony

with Definition 1.8 and the equivalence established in Theorem 1.10 as

ωp(v1, . . . , vk) =

∑1≤j1<···<jk≤n

αj1,...,jk(p)(dxj1)p(v1) ∧ · · · ∧ (dxjk)p(v

k)

=∑

1≤j1<···<jk≤n

αj1,...,jk(p)(v1j1

) . . . (vkjk).

Definition 1.13. [1] For a k-form ω and an i-form η on an open subset U ⊂ Rn,the exterior product of forms ∧ is the defined so that ω ∧ η is the (k+ i)-form givenby

(ω ∧ η)p = ωp ∧ ηp for all p ∈ U.

In order to make the equations to come less cluttered, we establish some notation

Notation : When a form is written in a basis (dxi)p the elements are always concate-nated by the wedge product and are understood to be with respect to the same pointp, we will further on leave out the wedge, and the indexing p, and (dxi)p ∧ (dxj)pwill be written as dxidxj. In this notation

ωp =∑

1≤j1<···<jk≤n

αj1,...,jk(p)dxj1 . . . dxjk .

Sometimes, for further abbreviation, put n = 1, . . . , n and, for any ordered indexset I, let Ik(I) denote the index set of all ordered sets of k elements

Ik(I) = j1, . . . , jk | j1 < · · · < jk, j1, . . . , jk ∈ I.

Then ω may also be written as

ω =∑

J∈Ik(n)

αJdxJ .

Definition 1.14. [1] A differentiable k-form is a k-form where all the coefficientsαj1,...,jk are differentiable functions. We denote the set of all differentiable k-forms inan open subset U of Rn by Ωk(U). It is then clear that if ω ∈ Ωk(U) and η ∈ Ωj(U)are differential forms in U then so is ω ∧ η ∈ Ωk+j(U).

A significant feature of these forms is how they interact with linear mappings ofthe space.

Definition 1.15. [3] A differentiable map f : Rn → Rm induces a map Ωk(Rm)→Ωk(Rn) denoted f ∗ such that for ω ∈ Ωk(Rm) we get f ∗ω ∈ Ωk(Rn), given by

(f ∗ω)p(v1, . . . , vk) = (ωf(p) dfp)(v1, . . . , vk) = ωf(p)(dfp(v1), . . . , dfp(vn)),

where dfp is the linear differential of the tangent space at p induced by f . We let f ∗

denote the associated map, for all degrees k, since which map to apply will always

7

be understood by the degree of the form ω. As convention, for α ∈ Ω0(Rn) a 0-form,also set

f ∗α = α f.

Proposition 1.16. [3] For a differentiable map f : Rn → Rm, ω, η ∈ Ωk(Rm) andα ∈ Ω0(Rm) we have

a) f ∗(ω + η) = f ∗ω + f ∗η,

b) f ∗(αω) = (f ∗α)(f ∗ω),

c) if ϕ1, . . . , ϕk ∈ Ω1(Rm), then f ∗(ϕ1 ∧ · · · ∧ ϕk) = f ∗(ϕ1) ∧ · · · ∧ f ∗(ϕk).Further

d) f ∗(ω ∧ µ) = (f ∗ω) ∧ (f ∗µ), µ ∈ Ωj(Rm)

e) (f g)∗ω = g∗(f ∗ω), for any differentiable map g.

We now prove a result that will be useful as reference later in chapter 4.

Theorem 1.17. For an injective differentiable map f : Rn → Rm (n ≤ m) and adifferentiable form ω ∈ Ωk(Rm), let η be the part of ω dual to the tangent vectors off(Rm), i.e.

ηp = ωp|∧k(Tpf(Rn))∗ .

Thenf ∗ω = f ∗η

The reader acquainted with Riemannian geometry, may see the corespondanceto how a submanifold with induced metric can be seen to inherit the Levi-Civitaconnection of the whole space by restriction of the connection to the tangential partwith respect to the submanifold [7].

Proof. When n = m we have that Tf(p)f(Rn) = Tf(p)Rm so that ωp|∧k(Tpf(Rn))∗ = ωpmand the result is trivial.When n < m an injective differentiable map f : Rn → Rm defines a submanifoldf(Rn) in Rm and at any point p ∈ Rn the differential dfp is a linear map fromTpRn to Tf(p)f(Rn). We also have that Tf(p)f(Rn) is a linear subspace of Tf(p)Rm.

For any 1-form ω ∈ Ω1(Rm) we have by definition ωf(p) ∈ ∧1(Tf(p)Rm)∗. ω can

thus be written as the sum of two 1-forms η, µ ∈ Ω1(Rm) such that for each pointf(p) ∈ f(Rn) η is contained in the dual of the tangent space of f(Rm) and µ in itsorthogonal complement. That is for all points f(p)

ωf(p) = ηf(p) + µf(p), where ηf(p) ∈∧1(Tf(p)f(Rn))∗, µf(p) ∈∧1

((Tf(p)f(Rn))⊥)∗.

Since both f and ω are differentiable, η and µ can be chosen to be so too. For anypoint p in Rn and any tangent vector vp ∈ TpRn we have

(f ∗ω)p(vp) = ωf(p)(dfp(vp)) = (η + µ)f(p)(dfp(vp)) = ηf(p)(dfp(vp)) + µf(p)(dfp(vp)).

But clearly by construction µf(p)(dfp(vp)) = 0 so that

(f ∗ω)p(vp) = ηf(p)(dfp(vp)) = (f ∗η)p(vp)

8

This holds for any choices of point p and vector vp , hence

f ∗ω = f ∗η.

We have thus proven the result for 1-forms, but since any k-form can be expressedas the exterior product of k 1-forms, statement c) in Theorem 1.16 ensures that thestatement also holds for a general k-form.

1.4 The Exterior Derivative

We have extensively used the differentiation of differentiable functions and shall nowextend this notion to k-forms in general. Let α ∈ Ω0(Rn) be a 0-form and then itsdifferential is a 1-form given by

dα =n∑i=1

∂α

∂xidxi,

with respect to the basis dx1, . . . , dxn[1, 3]. We will now generalise this to formsof higher degree.

Definition 1.18. [3] The exterior derivative d : Ωk(Rn) → Ωk+1(Rn) is defined sothat for any k-form

ω =∑

J∈Ik(n)

αJdxJ ∈ Ωk(Rn)

we have

dω =∑

J∈Ik(n)

dαJ ∧ dxJ =∑

J∈Ik(n)

n∑i=1

∂αJ∂xi

dxi ∧ dxJ .

Proposition 1.19. [1, 2, 3] Let α ∈ Ω0(Rn), ω, η ∈ Ωk(Rn) and µ ∈ Ωj(Rn). Thenthe exterior derivative as defined above has the following properties:

a) dα =∑n

i=1∂α∂xidxi,

b) d(ω + η) = dω + dη,

c) d(ω ∧ µ) = dω ∧ µ+ (−1)kω ∧ dµ,

d) d(dω) = d2ω = 0.

Further, for a differentiable function f : Rl → Rn we have

e) d(f ∗ω) = f ∗(dω).

By the following we make sure that the operator is independent of the choice ofbasis.

Theorem 1.20. [2] There exists a unique operator d : Ωk(Rn) → Ωk+1(Rn) withproperties a) - d) of Proposition 1.19.

We have now fully described the k-forms and exterior differentiation of theseobjects. In order to properly define integration of forms we need to define how todescribe and handle the region to integrate over. That will be the attention of thenext chapter.

9

10

Chapter 2

Integration of Forms, SingularCubes and Chains

In this chapter we define integration in the language of forms. We will start offby defining integration over the simplest regions, to continue with integration oversimply connected regions and a sum of these. This leads up to the possibility to stateand prove Stokes’s general theorem for these type of regions. In the next chapterthis will play a fundamental role in describing integration and Stokes’s theorem onmanifolds with boundary.

2.1 Integration on Singular Cubes and Chains

Well known integration in analysis of several variables would be written as∫[0,1]n

f(x1, . . . , xn)dx1 . . . dxn

and can be calculated by iteration (Fubini’s theorem)[1],∫ 1

0

(. . . (

∫ 1

0

f(xj1 , . . . , xjk)dx1) . . . )dxk.

The language we have established for differential forms, is developed so that thisrelation still holds.

Definition 2.1. [1] Let ω ∈ Ωn(U) be a differential k-form on an open set U ∈ Rn

containing the unit cube [0, 1]n ⊂ U . let ω be expressed in the canonical basis sothat ω = α · dx1 ∧ · · · ∧ dxn, where α : U → R is uniquely determined by ω. Thenwe define the integral of ω over [0, 1]n by∫

[0,1]nω =

∫[0,1]n

α(x1, . . . , xn)dx1 . . . dxn.

Since a mapping of [0, 1]n into a space induces a reverse mapping of forms, wemay use this simple definition to define integration of forms in any area that canbe mapped to bijectively by the standard cube. These maps will be called singularcubes.

Definition 2.2. [1] A singular k-cube in a region R of Rn is a continuous bijectivefunction c : [0, 1]k → R.

11

If ω is a differentiable form in the same region R of Rn, then c∗ω is a form in[0, 1]k, only differentiable if c also is. Since our interest in cubes is in mapping formsand maintaining their differentiability, we will limit our attention to this case, sothat all singular cubes henceforth will be assumed to be differentiable. Let ck(R)denote the set of all differentiable singular k-cubes in R.Since the composition of differentiable functions is differentiable it is clear that for ak-cube c ∈ ck(R) and a j-cube d ∈ cj([0, 1]k) the composition is also a differentiablej-cube c d ∈ cj(R)

Definition 2.3. [1] The standard k-cube is the inclusion map Ik : [0, 1]k → Rk

defined byIk(x) = x

It is clear that Ik ∈ ck(Rk).

Definition 2.4. [1] Let k be a positive integer and ω ∈ Ωk(R), c ∈ ck(R). Thendefine ∫

c

ω =

∫[0,1]k

c∗ω.

For k = 0 and ωp = α(p) we define∫c

ω = α(c(0)).

We have now defined integration for any area that can be mapped to by a singlesingular cube. In order to integrate structures that can be described as a sum ofsingular cubes, we need to construct an object to handle this.

Definition 2.5. [1] A singular k-chain in R is a function f : ck(R)→ Z with finitesupport (that is f(c) 6= 0 for only finitely many c ∈ ck(R)). Let Ck(R) denote theset of all singular k-chains in R.

Definition 2.6. [1] For the set Ck(R) of singular k-chains in R, we define an additionand a multiplication by integers such that for all c ∈ ck(R), f, g ∈ Ck(R) and i ∈ Zwe have

(f + g)(c) = f(c) + g(c),

(if)(c) = i(f(c)).

With this definition it is clear that Ck(R) is closed under addition and multipli-cation by integers. Now for every c ∈ ck(R) there is a dual function in Ck(R) whichwe denote by c, such that c(c) = 1 and for c′ 6= c, c(c′) = 0

Theorem 2.7. [1] Every singular k-chain s ∈ Ck(R) can be written as a finite sum∑aici, where ai ∈ Z and ci ∈ Ck(R) is the dual of ci ∈ ck(R). Further, if requiring

the chain to be chosen such that ci 6= cj for all i, j, then the representation is uniqueup to ordering.

The purpose of introducing chains is to give us the possibility to define integrationover a sum of cubes, and thus we have the following definition.

Definition 2.8. [1] For ω ∈ Ωk(R) and s ∈ Ck(R), written as s =∑aici where

ai ∈ Z and ci ∈ ck(R) define ∫s

ω =∑

ai

∫ci

ω.

12

Given a k-cube c ∈ ck(R) we may by a change of orientation, ζi or ξk,j : [0, 1]k →[0, 1]k where

ζi(x1, . . . , xi, . . . , xk) = (x1, . . . , 1− xi, . . . , xk)

ξi,j(x1, . . . , xi, . . . , xj, . . . , xk) = (x1, . . . , xj, . . . , xi, . . . , xk)

construct other k-cubes

cζi = c ζi

cξi,j = c ξi,j.

But when integrating any n-form ω over these cubes, we note that∫cζi

ω =

∫cξk,j

ω = −∫c

ω =

∫−cω.

Since our focus defining chains in this paper is for the use in treating integration,the chains cζi , cξi,j , and −c will be considered equivalent, and to multiply any chains with −1 to −s may be referred to as a change of orientation.

2.2 The Boundary Operator

We are now ready to introduce the boundary operator on singular chains, denotedby ∂, which maps an k-chain onto an (k-1)-chain called its boundary. In order to doso, we start by finding the notion of the boundary of a simple cube, correspondingto the intuitive notion. And we start of with the standard k-cube Ik.

Definition 2.9. [1] For i ∈ k and δ ∈ 0, 1 we define the singular (k-1)-cubes Ik(i,δ)so that for each x ∈ [0, 1]k−1

Ik(i,δ)(x) = Ik(x1, . . . , xi−1, δ, xi+1, . . . , xn)

These are called the (i, δ)− faces of Ik.

Definition 2.10. [1] For a general k-cube c ∈ ck(R) we define its (i, δ)-face c(i,δ) ∈ck−1(R) by

c(i,δ) = c Ik(i,δ).

For a differentiable singular n-cube c and ∂ denoting the topological boundary ofa set, it is clear that the image of the boundary of the region [0, 1]k, is the union ofthe images of the cubes faces, that is

c(∂[0, 1]k) =k⋃i=1

⋃δ∈0,1

c(i,δ)([0, 1]n−1).

This, together with a goal of counter clockwise induced orientation motivates thefollowing definition of the boundary operator on chains.

13

Definition 2.11. [1] We define for each chain c ∈ Ck(R) the dual of a cube c ∈ ck(R)its boundary with preserved orientation ∂c ∈ Ck−1(R) as

∂c =k∑i=1

∑δ∈0,1

(−1)i+δc(i,δ).

that is, the sum of the duals of the cubes faces with a sign for orientation. Wedefine the boundary operator ∂ : Ck(R) → Ck−1(R) for a general singular k-chains =

∑aici by

∂s = ∂(∑

aici) =∑

ai(∂ci).

Theorem 2.12. [1, 2] For all s ∈ C the boundary operator ∂ satisfies

∂2s = ∂(∂s) = 0.

2.3 The General Theorem of Stokes on Singular Chains

We are now ready to state and prove Stokes’ general theorem on singular chains.

Theorem 2.13 (The general theorem of Stokes on singular chains). [1] For a sin-gular k-chain s ∈ Ck(U) and a differential (k−1)-form ω ∈ Ωk−1(U) on an open setU ⊂ Rn the following integral relation holds∫

s

dω =

∫∂s

ω.

Proof. For a k-1 form ω and a k-chain s =∑aici we have∫

s

dω =∑

ai

∫ci

dω =∑

ai

∫Ikc∗i (dω) =

∑ai

∫Ikd(c∗iω). (2.1)

similarly, by first establishing the equivalence for a singular cube c [1]∫∂c

ω =

∫∂Ik

c∗ω

we get ∫∂s

ω =∑

ai

∫∂ci

ω =∑

ai

∫∂Ik

c∗iω (2.2)

Thus, to prove the equality it suffices to show that∫Ikdη =

∫∂Ik

η, (2.3)

for η ∈ Ωk−1(V ), where v ⊂ Rn is an open set containing [0, 1]k, since this equates2.1 and 2.2. This equality will be easily established, due to the fact that the bound-ary operator applied on simple cubes, gives the signed endpoints, and so does thefundamental theorem of calculus together with the partial derivatives that come asa consequence of the exterior derivatives of forms.

14

We may write ω in the dual of the basis of Ik so that

ω =k∑j=1

αjdx1 . . . dxj . . . dxk,

where dxj is the omitted term. Thus equation 2.3 can be expressed in this basis as∫Ikdω =

k∑j=1

∫Ikdαjdx1 . . . dxj . . . dxk =

k∑j=1

∫∂Ik

αjdx1 . . . dxj . . . dxk =

∫∂Ik

ω.

This implies that we only need to consider the simplest form i.e. one of the elements

αjdx1 . . . dxj . . . dxk, separately [1, 2]. Thus all we need to prove is that for anyj ∈ k ∫

Ikdαjdx1 . . . dxj . . . dxk =

∫∂Ik

αjdx1 . . . dxj . . . dxk. (2.4)

For the boundary operator we have

∂Ik =k∑i=1

(−1)i(Ik(i,0) − Ik(i,1)).

This means that for the right hand side of equation 2.4,∫∂Ik

αjdx1 . . . dxj . . . dxk =k∑i=1

(−1)i(∫

Ik(i,0)

αjdx1 . . . dxj . . . dxk−∫Ik(i,1)

αjdx1 . . . dxj . . . dxk

)and for each (i, δ)-face Ik(i,δ)∫

Ik(i,δ)

αjdx1 . . . dxj . . . dxk =

∫[0,1]k−1

(Ik(i,δ))∗(αjdx1 . . . dxj . . . dxk).

The star operator ∗ is defined so that for each point p ∈ [0, 1]k−1

((Ik(i,δ))∗(αjdx1 . . . dxj . . . dxk))p = (ajIk(i,δ))p

(((Ik(i,δ))

∗dx1)p∧. . . dxj · · ·∧((Ik(i,δ))∗dxk)p

).

We have for the convoluted function

(aj Ik(i,δ)) = aj(x1, . . . , xj−1, δ, xj+1, . . . , xk).

For each dxl , where l ∈ (k\j), we have

((Ik(i,δ))∗dxl)p = (dxl)Ik

(i,δ)(p) (dIk(i,δ))p.

But for the given basisdIk(i,δ) = (1, . . . , 1, 0, 1, . . . , 1)

where 0 is in the i:th place. Thus

(Ik(i,δ))∗dxl =

0 if i = l

dxl if i 6= l

15

so that if i 6= j

(aj Ik(i,δ))p(((Ik(i,δ))

∗dx1)p ∧ . . . dxj · · · ∧ ((Ik(i,δ))∗dxk)p

)= 0.

Thus ∫[0,1]k−1

(Ik(i,δ))∗(αjdx1 . . . dxj . . . dxk) =

=

0 if i 6= j∫[0,1]k−1

αj(x1, . . . , xj−1, δ, xj+1, . . . , xk)dx1 . . . dxj . . . dxk if i = j

and therefore∫∂Ik

αjdx1 . . . dxj . . . dxk = (−1)j+1

∫[0,1]k−1

(αj(x1, . . . , xj−1, 1, xj+1, . . . , xk)

−αj(x1, . . . , xj−1, 0, xj+1, . . . , xk))dx1 . . . dxj . . . dxk.

(2.5)

On the other hand for the left hand side of equation 2.4∫Ikdαjdx1 . . . dxj . . . dxk =

∫Ik

n∑i=1

∂αj

∂xidxi ∧ dx1 . . . dxj . . . dxk

=

∫Ik

∂αj

∂xjdxj ∧ dx1 . . . dxj . . . dxk

= (−1)j−1

∫Ik

∂αj

∂xjdx1 . . . dxk.

From Fubini’s theorem and the fundamental theorem of calculus, we now get

= (−1)j−1

∫ 1

0

· · ·∫ 1

0

( ∫ 1

0

∂αj

∂xjdxj)dx1 . . . dxj . . . dxk

= (−1)j−1

∫ 1

0

· · ·∫ 1

0

([αj]10

)dx1 . . . dxj . . . dxk

= (−1)j+1

∫[0,1]k−1

(αj(x1, . . . , xj−1, 1, xj+1, . . . , xk)−

αj(x1, . . . , xj−1, 0, xj+1, . . . , xk))dx1 . . . dxj . . . dxk.

(2.6)

Now since expressions 2.6 and 2.5 are equal, then equation 2.4 is proven to hold.this proves the result.

16

Chapter 3

Manifolds

In previous chapter we introduced singular cubes and chains, defined the boundaryoperator and integral of forms on them, and proved Stokes’ General Theorem in thissetting. The goal of this chapter is to use these definitions and results to generalisethe theorem to compact manifolds with boundary. In order to do so, we first needto make clear what we mean by a manifold with boundary.

3.1 Definitions and Well Known Results

Definition 3.1. [3] For n ≥ 1 we define the n-dimensional upper half-space Hn by

Hn = x ∈ Rn : x1 ≥ 0

and equip this with the standard subset topology.

The definition of an n-dimensional differentiable manifold with boundary is al-most the same as for a manifold without, just that the local charts are defined fromopen sets of the half-space Hn.

Definition 3.2. [3] An m-dimensional differentiable manifold with a regular bound-ary is a set M and a collection of local charts (Uα, fα), where Uα are open subsetsof Hm and fα are injective maps fα : Uα →M , satisfying

1.⋃α fα(Uα) = M,

2. for all pairs α, β, if the intersection of their images W = fα(Uα) ∩ fβ(Uβ) isnon-empty, we have that the sets f−1

α (W ), f−1β (W ) are open in Hm and the

transition maps f−1α fβ, f−1

β fα are differentiable.

3. The family (Uα, fα) is maximal with respect to (1) and (2), that is it in-cludes all local chart (Uβ, fβ) of M that satisfies (2) together with all chart of(Uα, fα)

The set (Uα, fα) is called a differentiable structure on M if it satisfies (1),(2) andit is considered maximal if it also satisfies (3).

The topology of the manifold is inherited by the charts so that the inverse map-pings f−1

α are continous. Since the collection Uα, fα satisfies (2), this is welldefined.

17

A point p ∈ M is said to be a boundary point of M if for some parametrisationfα : Uα →M around p we have that fα(0, x2, . . . , xm) = p. The set of all boundarypoints is called the boundary of M and is denoted ∂M [3].

Note that if for any two parametrisations (Uα, fα), (Uβ, fβ) and a point x =(0, x2, . . . , xm) ∈ Uα such that fα(x) ∈ fβ(Uβ) then as a consequence of property (2)of Definition 3.2, we have

f−1β fα(x) = (0, y2, . . . , ym) yi ∈ R.

Hence the definition of the boundary is independent of parametrization. Thus theboundary is well defined [3]. Bear in mind that this definition of the boundary ofa manifold is not equivalent to the standard definition of the boundary of a set. Ifthe manifold is closed however, the definitions coincide[1].

We may also ask how this corresponds to the definition of the boundary of singularchains and cubes. Since a differentiable singular m-cube c in a manifold M togetherwith any local chart (Uα, fα) satisfies the requirements of property (2) of Defenition3.2, we may draw the following conclusion.

Theorem 3.3. For a singular m-cube c on an m-dimensional manifold M the bound-ary of the manifold ∂M within the image of the cube, is also within the image of itstopological boundary ∂c, that is

c([0, 1]m) ∩ ∂M ⊂ ∂c([0, 1]m) = c(∂[0, 1]m)

Proposition 3.4. [3] For an m-dimensional differentiable manifold M with regu-lar boundary satisfying definition 3.2, the boundary ∂M is an (m-1)-differentiablemanifold.

We will also need the notion of orientation.

Definition 3.5. [3] Let M be a differentiable manifold with boundary. For localparametrisations fα, fβ if they are (directly) comparable, that is fα(Uα)∩fβ(Uβ) 6= ∅,they are said to have the same orientation if the change of coordinate map f−1

α fβ :Rm → Rm preserves orientation. That is det(d(f−1

α fβ)) > 0. If for M there isa differentiable structure Uα, fα were all (directly) comparable parametrisationshave the same orientation, we say that M is orientable, otherwise it is not orientableor a non-orientable manifold. The classical example of a non-orientable manifold isthe Mobius strip [1, 6].

Proposition 3.6. [3] If M is orientable its orientation induces an orientation onits boundary ∂M.

3.2 Manifolds and Forms

In Section 1.3 we defined the differential forms in Rn. We now have a similardefinition for the differential forms on manifolds.

Definition 3.7. [1] For an m-dimensional differentiable manifold M with boundary

∂M , let p be a point in M . An element λp ∈ ∧k(TpM)∗ is called an (exterior)

k-form at p. For an open subset U of M a k-form in U is a map ω which associates

to each point p ∈ U a k-form ωp ∈∧k(TpRm)∗ at p. Given a local parametrisation

18

fα : Uα→M we define the local representation of ω in fα(Uα) with respect to thisparametrisation, as the exterior k-form ωα in Uα ⊂ Hm ⊂ Rm given by

ωαp = f ∗αωp.

Proposition 3.8. [2] Let ω be a k-form on a differentiable manifold M. Given alocal parametrisation fα : Uα → M let ωα be a local representation of ω in Uα. Ifωα is differentiable in Uα, then ω is differentiable in fα(Uα) in any representation.

Proof. Let ωα be a differentiable local representation of ω in fα(Uα). If fα(Uα) ∩fβ(Uβ) = W is non empty, then we have for a point p ∈ W with a change ofcoordinates and by Proposition 1.16

(f−1α fβ)∗ωαp = (f ∗β (f−1

α )∗)(f ∗αωp)

= (f ∗β (f−1α )∗ f ∗α)(ωp)

= (f ∗β (fα f−1α )∗)(ωp)

= f ∗βωp = ωβp .

By definition f−1α fβ is differentiable, so if ωα is differentiable in f−1

α (W ), thenωβ is differentiable in f−1

β (W ).

Definition 3.9. [1] For a differentiable manifold M with boundary ∂M a differentialk-form ω in a subset U of M is an exterior k-form such that for each p ∈ U there issome representation ωα, where p ∈ fα(Uα) such that ωα is differentiable in Uα.

Theorem 3.10. [1] For a p-form ω on a manifold M there is a unique (p+ 1)-formdω on M such that for every parametrisation fα : Uα →M we have the equality

f ∗α(dω) = d(f ∗αω).

This defines the differential operator for forms on manifolds.

Definition 3.11. For two manifolds M,N and a differentiable mapping φ : M → N ,we define φ∗ so that for any local parametrisation (Uα, fα) of M we have

f ∗ φ∗ = (f φ)∗

3.3 Integration on Manifolds

In order to integrate on a manifold M we will need it to be Hausdorff, orientableand to have a countable basis, see [3]. This will therefore be assumed from here on.

Definition 3.12 (Partition of unity). [9] For a manifold M with an indexed opencovering Vα. An indexed family of differentiable functions Ψ = ψβ where

ψα : M → [0, 1]

is said to be a differentiable partition of unity on M dominated by Vα, if:

1) ψα is only non-zero within Vα,

2) for each p ∈M only a finite number of functions ψ ∈ Ψ are non-zero,

3) for each p ∈M we have∑

Ψ

ψ(p) = 1.

Due to the restriction of property (2) the sum in property (3) is always well defined.

19

We now state the following.

Theorem 3.13 (the existence of partition of unity). [1] For any open cover Vαof a manifold M there exists a differentiable partition of unity Ψ dominated by it.

Theorem 3.14. For an m-dimensional manifold with boundary M defined by thecollection (Uα, fα). Let V be the collection of sets Vα = fα(Uα) Then there is acountable collection of singular m-cubes C = ck in M such that the collection oftheir images P = ck([0, 1]m) together with the collection of its interiors P =P k |Pk ∈ P satisfies that P is a refinement of V and P is an open cover ofM.

This means that we may cover any manifold M with singular cubes, and stillensure that each cube is contained within the image of a local chart. Further thata partition of unity may be constructed dominated by the collection of the inte-rior of the simple cubes domains. The proof requires a bit of insight in topology,and only an outline will be given. One way is to first find a countable open coverWk such that for each Wi there is some local chart (Uα, fα)i such that the preim-age Wi = f−1

αi(Wi) is an open hyper-rectangle in Hm. Then let Ki be the subset

of M uniquely covered by Vi, that is Ki = M\(⋂j 6=iWj). We then find that its

preimage Ki = f−αi1(Ki) is a compact subset of the open set Wi, and thus has anon-zero distance to its boundary in the topology of Hm. This makes it easy tofind a singular m-cube ci ∈ C(Hm) such that for its image Pi = ci([0, 1]m) we haveKi ⊂ P i ⊂ Pi ⊂ Wi. Then the collection of singular m-cubes fαi ci satisfies thetheorem.

We can now start defining integration on manifolds.

Definition 3.15 (integration on a cube in a manifold). [1] For a k-form ω on anm-dimensional manifold M with boundary ∂M and a singular k-cube c on M, wedefine ∫

c

ω =

∫[0,1]kc∗ω.

Theorem 3.16. [1] Let M be an m-dimensional oriented manifold, c1, c2 : [0, 1]m →M be two orientation-preserving m-cubes and ω be an m-form on M such that

ω(p) = 0 whenever p /∈ c1([0, 1]m) ∩ c2([0, 1]m).

Then ∫c1

ω =

∫c2

ω

Definition 3.17. [1] Let M be an oriented m-dimensional manifold, ω be an m-formon M and c be an orientation-preserving singular m-cube c in M. If ω has its supportwithin the image of c, then we define∫

M

ω =

∫c

ω.

By Theorem 3.16 this is well defined and does not depend on the choice of c.

This together with a partition of unity is enough to define integration of anym-form on the whole manifold.

20

Definition 3.18. For a differentiable m-form ω on an orientable m-dimensionalmanifold M, there is an open covering Vα dominated by a set of orientation pre-serving singular m-cubes cα, that is

Vα ⊂ cα([0, 1]m).

The existence of such a covering is established by Theorem 3.14. Let Ψ be a differ-entiable partition of unity dominated by Vα, then we define∫

M

ω =∑ψ∈Ψ

∫M

ψ · ω

if it converges.

Theorem 3.19. [1]The integral∫Mω in Definition 3.18 is independent of the choice

of covering and the partition of unity. Further, if M is compact, they can be chosenso that Ψ is finite.

We can hence state a couple of results concerning mappings of manifolds andtheir induced maps of forms in relation to integration.

Theorem 3.20. For two manifolds M, N and a differentiable mapping φ : N →M ,we have ∫

φ(N)

ω =

∫N

φ∗ω

This result can be established by studying the mapping of local parametrisationsof N and the composition rule of Definition 3.11.

Next result follows from Theorem 1.17 and shows that for a submanifold, onlythe tangential part of a differential form contributes to the integral over it.

Theorem 3.21. For a differentiable manifold M , an embedded k-dimesional sub-manifold N , and a differentiable form ω ∈ Ωk(M), let η be the part of ω dual to thetangent vectors of N , that is

ηp = ωp|∧k(TpN)∗ .

Then ∫N

ω =

∫N

η

Proof. let φ : N → M be the identity map, then φ(N) = N , and for η defined asabove, by Theorem 1.17 and Theorem 3.20∫

N

ω =

∫φ(N)

ω =

∫N

φ∗ω =

∫N

φ∗η =

∫φ(N)

η =

∫N

η

3.4 The General Theorem of Stokes on Manifolds With Bound-ary

With Stokes’ Theorem on singular chains proven, and integration of forms on man-ifolds defined using chains in local coordinates, we are now ready to stitch thistogether to prove the main theorem of this paper.

21

Theorem 3.22 (The General Theorem of Stokes on Manifolds With Boundary).[1, 2, 3] For a compact oriented m-dimensional manifold M with boundary ∂M anda differentiable (m− 1)-form ω on M, we have the following integral identity:∫

M

dω =

∫∂M

ω

where the boundary ∂M has the induced orientation.

Proof. [1] We first observe the result for two special cases, and then show how thegeneral case can be analysed with these in mind.Case 1 There is an orientation-preserving m-cube c in M \ ∂M such that ω = 0outside of c([0, 1]m). Then By Stokes’ theorem on chains (Theorem 2.13 )∫

c

dω =

∫∂c

ω

Now, since ω = 0 on ∂M and by continuity also on ∂(c([0, 1]m)), we have∫M

dω =

∫c

dω =

∫∂c

ω = 0 =

∫∂M

ω.

Case 2 There is an orientation preserving singular m-cube c in M such that c(k,α)

is the only face in ∂M and ω = 0 outside of c([0, 1]m). Then∫M

dω =

∫c

dω =

∫∂c

ω =

∫∂M

ω.

Now in the general case, there is an open cover Vα of M and a dominatedpartition of unity Ψ such that for each ψ ∈ Ψ the form ψ · ω is of one of the twopreviously considered cases. Ψ is a partition of unity so that for any point p in M∑

Ψ

ψ(p) = 1.

It directly follows that

0 = d(1) = d(∑

Ψ

ψ(p))

=∑

Ψ

dψ(p)

and thus ∑Ψ

dψ ∧ ω = 0.

Since M is compact, the sum is finite and∑Ψ

∫M

dψ ∧ ω = 0.

Therefore∫M

dω =∑

Ψ

∫M

ψ · dω =∑

Ψ

∫M

(dψ ∧ ω + ψ ∧ dω

)=∑

Ψ

∫M

d(ψ · ω) =

= [as a sum of forms of case 1 and case 2] =∑

Ψ

∫∂M

ψ · ω =

∫∂M

ω.

22

We have now proven the main theorem of this paper and as foreshadowed mostof the work was in defining the appearing terms properly. In the next chapter willshow how the General theorem of Stokes can be used as a key tool to prove otherresults.

23

24

Chapter 4

Applications

In this chapter we show how the general Theorem of Stokes can be used to proveother interesting results. We start by showing how the classical theorems of vectorcalculus follow as special cases. Then we discuss some other results on compact man-ifolds without boundary where The general Theorem of Stokes is instrumental. Thisstudy will mostly concern hypersurfaces and functions of their principal curvatures.This last section requires deeper acquaintance with Riemannian geometry, and willwithout specific definitions use notions and notation as given in Gudmundsson’sscript [7].

4.1 Classical Theorems

In vector calculus there are three major operators: the gradient, the rotation andthe divergence. In the language of forms and exterior derivatives, these are directlyconnected. First recall that

R3 ∼= (R3)∗ =∧1(R3)∗

by the canonical map χ :∧1(R3)∗ → R3 defined in a basis x1, x2, x3 of R by

χ(a1dx1 + x2dx2 + a3dx3) = (a1, a2, a3).

This means that for any 1-form ω we may define a vector field f : R3 → R3 by

f(p) = χ(ωp). (4.1)

In the rest of this text we will say for any vector field f and 1-form ω satisfyingequality 4.1 that f corresponds to ω. We will also note it as f ≡ ω or ω ≡ f .

We start with a 0-form i.e. a differentiable function

f : R3 → R

Then the exterior derivative of f corresponds to the gradient of f , that is

df =∂f

∂x1

dx1 +∂f

∂x2

dx2 +∂f

∂x3

dx3 ≡( ∂f∂x1

,∂f

∂x2

,∂f

∂x3

)= grad(f).[1, 3]

Remember that the dimension of the space of k-vectors in Rn is(nk

), so that the

space of k-vectors and the set of (n−k)-vectors have the same dimension and hence

25

they are isomorphic. This also means that the vector space of k-forms is isomorphicto the vector space of (n−k)-forms. The canonical isomorphism is called the Hodgestar operator.

Definition 4.1. [2, 3] For a k-form ω ∈ Ωk(Rn) we define the linear Hodge staroperator

? : Ωk(Rn)→ Ω(n−k)(Rn)

by setting?dxI = (−1)σdx(n\I)

where I ∈ Ik(n), and σ is 0 or 1 depending on if (I, (n\I)) is an even or oddpermutation of (1, . . . , n).

Note that for any I ∈ Ik(n) the permutation of (I, (n\I)) to ((n\I), I) is oddif and only if both k and (n-k) are odd. Hence ? ? ω = (−1)k(n−k)ω so if n is odd,? ? ω = ω [2, 3].Every linear mapping is fully described by how it acts on a basis. For the 1-formsdx1, dx2, dx3 ∈ Ω1(R3) forming an orthonormal basis, we have

?dx1 = dx2dx3, ?dx2 = −dx1dx3, ?dx3 = dx1dx2. [3]

If we start with a 1-form ω i.e the correspondence of a differentiable vector field

ω = a1dx1 + a2dx2 + a3dx3.

This yields its exterior derivative

dω =(∂a3

∂x2

− ∂a2

∂x3

)dx2dx3 +

(∂a3

∂x1

− ∂a1

∂x3

)dx1dx3 +

(∂a2

∂x1

− ∂a1

∂x2

)dx1dx2.

Applying the Hodge star operator gives the analogous result for the rotation operator

?dω =(∂a3

∂x2

− ∂a2

∂x3

)dx1 +

(∂a1

∂x3

− ∂a3

∂x1

)dx2 +

(∂a2

∂x1

− ∂a1

∂x2

)dx3. ≡ rot(χ(ω))

The divergence has the corresponding form

div(χ(ω)) ≡ ?d ? ω =∂a1

∂x1

dx1 +∂a2

∂x2

dx2 +∂a3

∂x3

dx3.

The above discussion gives us a way to show some well known results.

Theorem 4.2. [10] For a differentiable function f : U ⊂ R3 → R and a vector fieldV : U ⊂ R3 → R3 the operators grad, rot and div have the following properties:

a) rot(grad(f)) = 0,

b) div(rot(V )) = 0,

c) rot(rot(V )) = grad(div(V ))− div(grad(V )),where for V = (v1, v2, v3), vi : R3 → R,div(grad(V )) =

(div(grad(v1)), div(grad(v2)), div(grad(v3))

).

Proof. Properties a) and b) follow by the correspondence to 1-forms and that forany form ω we have ? ? ω = ω and ddω = 0. Property c) can be found to be true bystarting out with a corresponding 1-form and sorting out the substitutions.

26

By the correspondence between 1-forms and vector fields we can now prove thefollowing classical Theorem.

Theorem 4.3 (Green’s Theorem). [1] Let M be a compact simply connected regionof R2. For two differentiable functions α, β : M → R we have∫

∂M

αdx1 + βdx2 =

∫M

( ∂β∂x1

− ∂α

∂x2

)dA,

where dA denotes the surface element.

Proof. Let the 1-form ω be defined by

ω = αdx1 + βdx2.

Then by exterior derivation, we have

dω =( ∂β∂x1

− ∂α

∂x2

)dA

and the result follows immediately from the general Theorem of Stokes.

Identifying the correspondence between the divergence of a vector field and theoperator ?d? of forms, we may prove the Divergence Theorem.

Theorem 4.4 (the Divergence Theorem). [3] Let M ⊂ R3 be a compact threedimensional manifold with regular boundary. For a differentiable vector field F onM, we have ∫

M

div(F )dV =

∫∂M

〈F,N〉dA.

Here dV denotes the volume element, dA the surface element with the induced ori-entation and N is the induced outer normal vector field to ∂M .

Proof. [3] Let ω be the 1-form on M dual to the vector field F. Then

?d ? ω = div(F ).

Since div(F ) is a differentiable map R3 to R we can considered it to be a 0-formand we have equality, so we may apply the Hodge star operator on both sides, andsince the dimension is odd

d ? ω = ? ? d ? ω = ?div(F ) = div(F )dV.

For an orientation of R3 let N be the induced outer normal vector field of ∂M .Consider, for a neighbourhood U ⊂ R3 of a point p in the boundary ∂M , a localbasis of orthogonal unit vector fields N,X1, X2 such that X1, X2 are tangent fieldsto ∂M . Then

(?ω)p(X1, X2) = ωp(N) ≡ 〈F,N〉pFurther at each such point p the vectors (X1, X2)p form a basis for the tangent spaceTp∂M . Thus by Theorem 3.21∫

∂M

?ω =

∫∂M

〈F,N〉dA,

27

and the result follows by Stokes’ Theorem∫M

d ? ω =

∫∂M

?ω.

corresponds to the equivalence∫M

div(F )dV =

∫∂M

〈F,N〉dA.

Theorem 4.5 (The Classical Stokes’ Theorem). [1] Let M ⊂ R3 be a compact ori-ented two-dimensional manifold with boundary ∂M and N be the unit outer normalvector field on M determined by the orientation of M. Let T be a unit vector fieldtangential to ∂M . Let F be a differentiable vector field on M. Then∫

M

〈rot(F ), N〉dA =

∫∂M

〈F, T 〉ds.

Proof. Let ω be the 1-form on M dual to the vector field F. At any point p in M wemay choose a local basis of orthogonal unit vector fields N,X1, X2 such that X1, X2

are tangent fields to M. Then

〈rot(F ), N〉p = (?dω(N))p = (dω(X1, X2))p.

Since (X1, X2)p form a basis for TpM we have by Theorem 3.21∫M

〈rot(F ), N〉dA =

∫M

dω.

Similarily, for the field T tangential to the boundary

〈F, T 〉p = (ω(T ))p

and Tp is a basis for the tangent space Tp∂M so that∫∂M

〈F, T 〉ds =

∫∂M

ω.

Thus by the General Stokes’ Theorem∫M

〈rot(F ), N〉dA =

∫M

dω =

∫∂M

ω =

∫∂M

〈F, T 〉ds

4.2 Hypersurfaces

The following results will apply to m-dimensional hypersurfaces immersed in theEuclidean space Rm+1 with the canonical topology. These have the convenience of aone dimensional normal to the tangent spaces. this can be used to define the shapeoperator of the surface, and further to study its principal curvatures. In order to doso we need some definitions and to establish some results. This section will followthe notation that is used in Gudmundssons script on Riemannian Geometry [7].

28

Definition 4.6. [7] Let M be a smooth manifold. A Riemannian metric g on Mis a tensor field g : C∞2 (TM)→ C∞0 (TM) such that for each p ∈ M the restrictiongp of g is an inner product on the tangent space TpM . The pair (M, g) is called aRiemannian manifold.

Definition 4.7. [5] An m-dimensional Riemannian manifold (M, g) together witha Riemannian immersion φ : (M, g) → (Rm+1, can) is called a hypersurface. Let Nbe a locally defined unit normal field on M, then the shape operator

S : C∞(TM)→ C∞(TM)

is locally defined as

Sp(Xp) = (∇XpN)p.

If the normal field is globally defined (that is if M is orientable), then so is the shapeoperator. A unit normal field N may be seen as a mapping from the hypersurfaceto the unit sphere N : M → Sm such that each point is mapped to the point on thesphere with the same unit normal, and may then be referred to as the Gauss mapof M [5].

Proposition 4.8. [6] The shape operator is symmetric and hence by the Spectraltheorem the eigenvalues for Sp are real.

Proof. Let X, Y be vector fields of M and N be a unit normal field, then

X(〈Y,N〉) = 0 = Y (〈X,N〉).

But

X(〈Y,N〉) = 〈∇XY,N〉+ 〈Y,∇XN〉 = 〈∇XY,N〉+ 〈Y, S(X)〉

and similarly

Y (〈X,N〉) = 〈∇YX,N〉+ 〈X,S(Y )〉.

Then

0 = X(〈Y,N〉)− Y (〈X,N〉)= 〈∇XY,N〉+ 〈Y, S(X)〉 − 〈∇YX,N〉 − 〈X,S(Y )〉= 〈∇XY −∇YX,N〉+ 〈Y, S(X)〉 − 〈X,S(Y )〉= 〈[X, Y ], N〉+ 〈Y, S(X)〉 − 〈X,S(Y )〉.

But [X, Y ] is a tangent field to M, so 〈[X, Y ], N〉 = 0 and therefore

〈Y, S(X)〉 = 〈X,S(Y )〉.

Lemma 4.9. [7] For an m-dimensional Riemannian manifold M and a point p ∈M , there exists a set of linearly independent vector fields X1, . . . , Xm defined in aneighbourhood of p such that for all i,j we have

∇XiXj −∇XjXi = [Xi, Xj] = 0.

29

Proof. Let (U, x) be local coordinates around the point p and let

Xi =∂

∂xi| i = 1, . . . ,m

be the induced local frame for TM. The map x : U → x(U) is bijective and thedifferential dx : TU → Tx(U) ∼= TRm is a Lie algebra homomorphism. The vectorfield Xk ∈ C∞(TU) is x-related to the coordinate vector field ∂ek ∈ C∞(Tx(U)) andthus for all f ∈ C2(x(U)) we have

dx([Xk, Xl])(f) = [∂ek , ∂el ](f) = ∂ek(∂el(f))− ∂el(∂ek(f)) = 0.

This shows that [Xk, Xl] = 0 so that all Xj commute, and since the Levi-Civitaconnection ∇ is torsion free, we have

∇XkXl −∇XlXk = [Xk, Xl] = 0.

Definition 4.10. [7] On a Riemannian manifold (M, g) the curvature tensor R :C∞3 (TM)→ C∞(TM) is defined by

R(X, Y )Z = ∇X∇YZ −∇Y∇XZ −∇[X,Y ]Z.

Lemma 4.11. For a hypersurface M in Rm+1 with shape operator S, defined on aneighbourhood Up of a point p, we have for commuting vector fields X, Y ∈ C∞(TUp)that

∇XS(Y ) = ∇Y S(X).

This result actually holds for general X, Y , not necessarily commuting, but theresult of Lemma 4.11 is all we will need, and this restriction makes the proof easier.

Proof. For a neighbourhood Up of p, choose two commuting tangent fields X,Y anda unit normal field N corresponding to the shape operator given by S(Z) = ∇ZN .Since Rn+1 is flat we have R(X, Y )N = 0 but by definition

R(X, Y )N = ∇X∇YN −∇Y∇XN +∇[X,Y ]N

and for X,Y commuting, ∇[X,Y ]N = ∇0N = 0. Thus

0 = ∇X∇YN −∇Y∇XN = ∇XS(Y )−∇Y S(X).

The value of a tensor field is independent of the choice of basis. We may docomputations in a suitable basis, as long as the resulting expressions are tensorial.In our case in the following sections this will be a basis of commuting vector fieldssince it significantly simplifies some proofs as shown above.

30

4.2.1 Isometric Ovaloids

Ovaloids are a special class of two-dimensional surfaces in R3.

Definition 4.12. A regular surface M in the Euclidean R3 is called an ovaloid if itis compact and the shape operator has strictly positive eigenvalues everywhere.

This class of surfaces is restricted enough to prove some quite substantial results.This section is mainly a reworking of Gudmundsson’s lecture notes on the subject[8].

To start of we prove some results also true for ovaloids higher dimensional coun-terparts.

Theorem 4.13. (Hadamard) [5] Let M be a connected closed m-dimensional hy-persurface immersed in Rm+1, where m > 1. If the shape operator is everywherepositive definite, then M is diffeomorphic to a sphere via the Gauss map. FurtherM is globally convex, that is each point p ∈M is the only point of M in the tangentplane TpM ⊂ Rm+1, so that M\p is contained strictly on one side of the tangentplane.

Proof. If the shape operator is strictly positive, then it is non-singular. Thus theGauss map N : M → Sm is a local diffeomorphism and by the closure of M itcovers Sm. If m > 1 then Sm is simply connected and therefore N must be adiffeomorphism.[5]

To assure the convexity we start by assuming it is not. Let p be a point in M suchthat M\p is not contained strictly on one side of TpM . Let Np be the a unit normalof M at p. Since M is diffeomorphic to a sphere it is bounded so that there aresupremum and infimum bounding hyperplanes PNpsup(M) and PNpinf (M) parallel toTpM so that M is contained in the half space below PNpsup(M) an above PNpinf (M)with respect to the normal direction Np. Inevitably these are tangent planes to Mso that for some points q, r ∈M we have TqM = PNpsup(M) and TrM = PNpinf (M).If TpM is one of these planes our assumption makes sure we still can choose q and rso that p, q, r are distinct points in M . Since the tangent planes are parallel, then soare the unit normals Np, Nq, Nr. Since there are always only two points on a spherewith parallel normals, the Gauss map of M can then not be injective contradicting itbeing a diffeomorphism, an hence M\p is contained strictly on one side of TpM .

Since a hypersurface M as in Theorem 4.13 is diffeomorphic to a sphere it sep-arates the space Rm+1 into a bounded interior and an unbounded exterior region.We may then refer to an outer normal at p as being the normal at p directed intothe outer region of M . An inner point q of M , being a point in the interior, has theproperty that any ray starting at q will intersect M at a finite distance from q. Theconvexity of M ensures that for any point p on M , an inner point p, and the outernormal Np we have

〈p− q,Np〉 > 0.

This since q is necessarily on the same side of TpM as M\p and Np is directed inthe opposite direction.

Ovaloids are quite a restricted subset of surfaces in R3. The main theorem of thissection proves that the metric of the ovaloid in great extent defines its immersion.

31

Theorem 4.14 (Isometric Ovaloids). [8] Two isometric ovaloids M,M∗ in R3 arecongruent i.e. one can be obtained from the other by an Euclidean motion.

In order to prove this we will compare two ovaloids M and M* in R3 using thefollowing notation. Let S and S* denote the corresponding shape operators andH, H* be their mean curvatures. For an isometric map φ : M → M∗ we defineSφ : TpM → TpM by

Sφp (X) = dφ−1(S∗φ(p)(dφX)).

Let Hφ be defined by2Hφ

p = trace(Sφp ).

We note that by the isometry,

det(Sφ) = det(dφ−1S∗dφ) = det(S∗) = det(S).

Also2Hφ

p = trace(Sφ) = trace(dφ−1S∗φ(p)dφ) = trace(S∗φ(p)) = 2H∗φ(p)

To prove Theorem 4.14 we wish to prove that S = Sφ. For that we will need thefollowing Theorem and results.

Theorem 4.15 (The Herglotz integral formula). [8] For two isometric compactsurfaces M and M∗ in R3, with shape operators S, S∗, mean curvatures H,H∗, andfor an isometric map φ : M → M∗ let Sφ and Hφ be defined as above. Then wehave

2

∫M

(H −Hφ)dA =

∫M

det(S − Sφ)〈N, p〉dA.

Proof. By Stokes’ Theorem and the compactness of M and M*, for a 1-form ω, wehave ∫

M

dω =

∫∂M

ω = 0.

If we can find a suitable 1-form ω such that

dωp = 2(H −Hφ)p − det(S − Sφ)p〈Np, p〉.

This would prove the result. The calculations will be done locally with an or-thonormal basis of vector fields. Choose two commuting differentiable vector fieldsX1, X2 tangent to M and a field N normal to M . At each point p, we have a localorthonormal basis N,X1, X2p. Then define the 1-form µ by

µp = det(p,Np, Sp(X1))dx1 + det(p,Np, Sp(X2))dx2.

Then

X1(det(p,Np, Sp(X2)) =det(∇X1p,Np, Sp(X2)) + det(p,∇X1Np, Sp(X2)) + det(p,Np,∇X1Sp(X2)) =

det(X1, N, S(X2))p + det(p, Sp(X1), Sp(X2)) + det(p,Np,∇X1Sp(X2)).

Similarly, we have

X2(det(p,Np, Sp(X1)) =det(∇X2p,Np, Sp(X1)) + det(p,∇X2Np, Sp(X1)) + det(p,Np,∇X2Sp(X1)) =

det(X2, N, S(X1))p + det(p, Sp(X2), Sp(X1)) + det(p,Np,∇X2Sp(X1)).

32

Thusdµp = det(X1, N, S(X2))p − det(X2, N, S(X1))p

+ det(p, Sp(X1), Sp(X2))− det(p, Sp(X2), Sp(X1))+ det(p,Np,∇X1Sp(X2))− (det(p,Np,∇X2Sp(X1))

= − det(N,X1, S(X2))p − det(N,S(X1), X2)p+ 2 det(p, S(X1), S(X2))+ det(p,Np, (∇X1Sp(X2)−∇X2Sp(X1))).

By Lemma 4.11 we have ∇X1Sp(X2) − ∇X2Sp(X1) = 0, so when described in thelocal basis we have

dµp = − det

1 0 00 1 sX1(X2)0 0 sX2(X2)

p

−det

1 0 00 sX1(X1) 00 sX2(X1) 1

p

+2 det

pN 0 0pX1 sX1(X1) sX1(X2)pX2 sX2(X1) sX2(X2

p

.

Thus

dµp = −2Hp + 2 det(Sp)〈Np, p〉.

This resulting expression is interesting and will be developed further in Theorem4.19.

Now considering Sφ, we find a similar form. We define

ηp = det(p,Np, S

φp (X1)

)dx1 + det

(p,Np, S

φp (X2)

)dx2

so that by a similar calculation [8] we yield

dηp =(− 2Hφ

p + det(p, Sp(X1), Sφp (X2))− det(p, Sp(X2), Sφp (X1))

+ det(p,Np,∇X1Sφp (X2)−∇X2S

φp (X1))

)dA.

In similarity of the calculation of dµ we hope that ∇X1Sφp (X2) − ∇X2S

φp (X1) = 0.

The Levi-Civita connection ∇ is only dependent on the metric and φ is isometric.This together with Theorem 4.11 yields

∇X1Sφp (X2)−∇X2S

φp (X1) = (φ−1)∗

(φ∗(∇X1S

φp (X2)−∇X2S

φp (X1)

))= (φ−1)∗

(∇dφ(X1)dφ(Sφp (X2))−∇dφ(X2)dφ(Sφp (X1))

)= (φ−1)∗

(∇dφ(X1)S

∗φ(p)(dφ(X2))−∇dφ(X2)S

∗φ(p)(dφ(X1))

)= (φ−1)∗

(0)

= 0.

Thus

dηp = −2Hφ+det

pN 0 0

pX1 sX1(X1) sφX1(X2)

pX2 sX2(X1) sφX2(X2

p

+det

pN 0 0

pX1 sφX1(X1) sX1(X2)

pX2 sφX2(X1) sX2(X2

p

.

By some (non trivial but easy to verify [8] ) substitutions, we get

dηp =(− 2Hφ +

(det(Sp) + det(Sφp )− det(Sp − Sφp )

)〈Np, p〉

)dA,

33

and since det(S) = det(Sφ)

d(ω − η)p =(2(Hφ

p −Hp) + det(Sp − Sφp )〈Np, p〉)dA

so that by Stokes’ theorem and the closure of M we finally reach at

2

∫M

(H −Hφ)dA =

∫M

det(S − Sφ)〈N, p〉dA.

Here we state and prove two well-known results from linear algebra.

Lemma 4.16. [8] Let A,B ∈ GLn(R) be two symmetric matrices. If A is positivedefinite there exist P,Λ ∈ GLn(R), such that Λ is diagonal,

PAP T = I and PBP T = Λ.

Proof. Since A is symmetric, all its eigenvalues are real, so it is diagonalisable, i.e.there exists an orthogonal matrix EA ∈ O(n) so that EAAE

TA = ΛA, where ΛA is of

the form

ΛA =

λA1 0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 λAn

.Since A is positive definite, all the eigenvalues are strictly positive, and we maydefine a real diagonal matrix VA, such that V 2

A = Λ−1A , that is

VA =

1√λA1

0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 1√

λAn

.Then we have

I = ΛAV2A = VAΛAVA = VAEAAE

TAVA = (VAEA)A(VAEA)T .

A symmetric transformation has a symmetric matrix as its representation in anyorthonormal basis, hence if the matrix B is symmetric, so is BA = EABE

TA. It is

also easy to see that also the matrix C = V BAV is symmetric. We may thus findan orthogonal matrix EC ∈ On(R) and a diagonal matrix ΛC such that

ECCETC = ΛC .

We can now define the sought after matrices P = ECV EA and Λ = ΛC . Then

PBP T = ECV EAB(ECV EA)T = ECV EABETAV E

TC = ECCE

TC = Λ

and

PAP T = ECV EAA(ECV EA)T = ECV EAAETAV E

TC = ECV ΛAV E

TC = ECIE

TC = I.

This is what we were looking for.

34

Lemma 4.17. [8] Let A,B ∈ GL2(R) be positive definite symmetric matrices. Ifdet(A) = det(B) then we have

det(A−B) ≤ 0,

with equality if and only if A = B.

Proof. Since A and B are positively definite and symmetric, we may find Λ and Pas in Lemma 4.16. Let λ1, λ2 be the eigenvalues of Λ. We then have

λ1λ2 = det(Λ) = det(P )2 det(B) = det(P )2 det(A) = det(I) = 1.

Since A and B are positive definite, so is Λ and therefore

0 ≥ (1− λ1)(1− λ2) = det(I − Λ) = det(PAP T − PΛP T )

= det(A−B) det(PP T )

=det(A−B)

det(A).

Since A is positive definite we obtain det(A−B) ≤ 0.If A = B it is clear det(A − B) = 0. If det(A − B) = 0 then (1 − λ1)(1 − λ2) = 0and thus λ1 or λ2 = 1 which implies that both are. Hence Λ = I and thereforeA = B.

We are now ready to prove Theorem 4.14.

Proof. (Theorem 4.14)[8] Choose coordinates in R3, so that 0 ∈ R3 is an innerpoint of M. Then by the convexity of M whe have that 〈Np, p〉 > 0, where p is anypoint on M and N is the outer normal field of M. Since M and M∗ are isometric,we have

det(S) = det(S∗) = det(Sφ)

and S, Sφ are both positive definite and symmetric. Therefore, by Lemma 4.17,

det(S − Sφ) ≤ 0.

so that by Herlotz integral formula∫M

(H −Hφ)dA ≤ 0.

By symmetry of these calculations, considering the opposite roles of M and φ(M) ⊂M∗ and the inverse mapping φ−1∫

φ(M)

(H∗ −Hφ−1

)dA ≤ 0.

But by sorting out the mappings∫φ(M)

(H∗ −Hφ−1

)dA =

∫M

φ∗((H∗ −Hφ−1

)dA).

But at each point p in M

(φ∗((H∗ −Hφ−1

)dA))p = (H∗φ(p) −Hφ−1

φ(p))φ∗dAp = (Hφ

p −Hp)φ∗dAp.

35

For any vectors v1, v2 ∈ TpM by definition φ∗dAp(v1, v2) = dAφ(p)(dφ(v1), dφ(v2))and since φ is an isometry, by definition dAφ(p)(dφ(v1), dφ(v2)) = dAp(v1, v2) [7],thus

(φ∗((H∗ −Hφ−1

)dA))p = ((Hφ −H)dA)p.

This means ∫M

(Hφ −H)dA ≤ 0

and thus also equal to zero so that again by Herlotz integral formula∫M

det(S − Sφ)〈N, p〉dA = 0

and since 〈N, p〉 > 0 we must have det(S − Sφ) = 0 and by Lemma 4.17 we getS = Sφ.

4.2.2 The k-Sectional Curvatures of Hypersurfaces

In the proof of Theorem 4.15 we found, for a three dimensional space, a 1-form µgiven by

µp = det(p,Np, Sp(X1))dx1 + det(p,Np, Sp(X2))dx2.

Differentiation yields

dµp = (−2Hp + 2 det(Sp)〈Np, p〉)dA.

If this is given for a compact surface M without boundary, we get directly by Stokes’Theorem ∫

M

dµ =

∫∂M

µ =

∫p

µ = 0.

Hence ∫M

(−2Hp + 2 det(Sp)〈Np, p〉)dAM = 0

so that ∫M

HpdAM =

∫M

det(Sp)〈Np, p〉dAM .

This result is very interesting, since it gives an integral relation between themean curvature and Gaussian curvature of a surface. The goal of this section is togeneralise this result for compact hypersurfaces. First we need a generalisation ofthe notion of mean curvature.

Definition 4.18. For a hypersurface M in Rm+1 with a locally defined shape oper-ator S, let λ1, . . . λm be the eigenvalues of Sp. Define the mean value of the productsof k of the eigenvalues of Sp by

Hk(Sp) =

(m

k

)−1∑1≤i1<···<ik≤m

λi1 · · ·λik .

Also set H0(Sp) = 1. We call this the k-sectional mean curvature of M at p.

We will now proceed with the goal of proving the following result, that relatesthese curvatures to each other.

36

Theorem 4.19. Let M be a compact hypersurface without boundary immersed inRm+1 with shape operator S = dN . Then we have the following identities∫

M

Hk(S)dVolM =

∫M

Hk+1(S)〈N, p〉dVolM , for k = 0, . . . , (m− 1).

In order to prove this, we need some more preparations. To get a better un-derstanding of how to prove Theorem 4.19 we will exemplify this by some specificresults. This to illustrate the general method.

Example 4.20. For a 2-dimensional manifold M immersed in R3, let ω : TpM → Rbe the 1-form defined for an orthogonal basis dx1, dx2 by

ω = α(X2)dx1 + α(X1)dx2,

with α(Xi) = det(p,N,Xi).

Then

dω =(X1α(x2)−X2α(x1)

)dx1 ∧ dx2

and

X1 det(p,N,X2)−X2 det(p,N,X1)

= det(X1, N,X2) + det(p, S(X1), X2) + det(p,N,∇X1X2)

−(det(X2, N,X1) + det(p, S(X2), X1) + det(p,N,∇X2X1))

= −2 det(N,X1, X2) + det(p, S(X1), X2)

+ det(p,X1, S(X2)) + det(p,N,∇X1X2 −∇X2X1).

But ∇X1X2 −∇X2X1 = [X1, X2] = 0. The remaining terms written explicitly interms of the basis N,X1, X2 give

−2 det

1 0 00 1 00 0 1

+ det

p1 0 0p2 s11 0p3 s12 1

+ det

p1 0 0p2 1 s21

p3 0 s22

= −2 + p1s11 + p1s22 = −2 + 〈p,N〉 trace(S).

Now, if the surface is orientable, compact and without boundary, we get by Stokes’theorem ∫

M

dω =

∫∂M

ω =

∫p

ω = 0.

It then follows that

2

∫M

dVolM =

∫M

trace(Sp)〈Np, p〉dVolM .

Also noted as ∫M

dVol =

∫M

Hp〈Np, p〉dVolM ,

where H is the mean curvature of M .

37

In general for an (m− 1)-form ω on a hypersuface M immersed in Rm+1

ω =m∑i=1

α(X1, . . . , Xi, . . . , Xm)dx1 . . . dxi . . . dxm

dωp =

(−

m∑i=1

(−1)iXi

(αp(X1, . . . , Xi, . . . , Xm)

))dVolp,

where α ∈ C1(Rm−1,R) and . denotes the omitted term.

To be able to look at this in another case, let us introduce some notation.

Definition 4.21. [4] For an n×n matrix A, we denote by [A]i the (n− 1)× (n− 1)principal minor determinant omitting the i-th row and i-th column of A. That is,for a nxn matrix

A =

a1,1 · · · a1,n...

. . ....

an,1 · · · an,n

we have the (n-1)x(n-1) matrix

[A]i =

a1,1 · · · a1,(i−1) a1,(i+1) · · · a1,n...

. . ....

.... . .

...a(i−1),1 · · · a(i−1),(i−1) a(i−1),(i+1) · · · a(i−1),n

a(i+1),1 · · · a(i+1),(i−1) a(i+1),(i+1) · · · a(i+1),n...

. . ....

.... . .

...an,1 · · · an,(i−1) an,(i+1) · · · an,n

Example 4.22. Let M be a compact hypersurface without boundary immersed inRm+1. We now consider the (m− 1)-form ω on M defined for a basis of commutingfields X1, . . . , Xm by

αp(X1, . . . , Xi, . . . , Xm) = det(p,Np, Sp(X1), . . . , Sp(Xi), . . . , Sp(Xm)

).

We then get

dωp =

(−

m∑i=1

(−1)iXi

(det(p,Np, Sp(X1), . . . , Sp(Xi), . . . , Sp(Xm))

))dVolp

and for each i we have

Xi

(det(p,Np, Sp(X1), . . . , Sp(Xi), . . . , Sp(Xm))

)= det(Xi, Np, Sp(X1), . . . , Sp(Xi), . . . , Sp(Xm))

+ det(p, Sp(Xi), Sp(X1), . . . , Sp(Xi), . . . , Sp(Xm))

+∑j 6=i

det(p,Np, Sp(X1), . . . , Sp(Xj−1), (∇XiS(Xj))p, Sp(Xj+1), . . . , Sp(Xi), . . . , Sp(Xm)

).

We can now look at each term and shift the Xi or Sp(Xi) into the i-th place. Thetwo first terms for each i give

(−1)i det(Np, Sp(X1), . . . , Sp(Xi−1), Xi, Sp(Xi+1), . . . , Sp(Xm))

38

+(−1)(i−1) det(p, Sp(X1), . . . , Sp(Xm))

= (−1)i(

[Sp]i − det(Sp)〈Np, p〉

).

To sort out the sum, we will consider the terms in all the sums for each i. In allthese sums, for any i 6= j there are only two terms where S(Xi) and S(Xj) are bothaltered or omitted at the same time, and no other term is altered. Without loss ofgenerality, assume j < i. Then these terms are

(−1)i det(p,Np, Sp(X1), . . . ,∇XiSp(Xj), . . . , Sp(Xi), . . . , Sp(Xm))

and(−1)j det(p,Np, Sp(X1), . . . , Sp(Xj), . . . ,∇XjSp(Xi), . . . , Sp(Xm))

Then the term ∇XiSp(Xj) will be as column (j + 2) of the determinant in the firstterm, and the term ∇XjSp(Xi) is found as column (i + 1) of the second. So, if weshift these term to the third column of each determinant, all other columns will beequal, thus shifting the terms j and (i− 1) columns respectively, and we get

(−1)(i+j) det(p,Np, (∇XiS(Xj))p, Sp(X1), . . . , Sp(Xj), . . . , Sp(Xi), . . . , Sp(Xm))

+ (−1)(j+i−1) det(p,Np, (∇XjS(Xi))p, Sp(X1), . . . , Sp(Xj), . . . , Sp(Xi), . . . , Sp(Xm))

= (−1)(i+j) det(p,Np, (∇XiS(Xj)−∇XjS(Xi))p, Sp(X1), . . . , Sp(Xj), . . . , Sp(Xi), . . . , Sp(Xm)

)By Lemma 4.11 we know that ∇XiS(Xj)−∇XjS(Xi) = 0. Therefore all these termscancel pairwise and we get

dωp = −m∑i=1

([Sp]i − det(Sp)〈Np, p〉)dVolp

=(m det(Sp)〈Np, p〉 −

m∑i=1

[Sp]i)dVolp.

By Stokes’ theorem, just as before this yields the equality∫M

m∑i=1

[S]idVolM = m

∫M

det(S)〈N, p〉dVolM .

These examples show the means of calculation that are needed to prove Theorem4.19, but in order to fully illuminate the connection, we need to give another defini-tion of the k-sectional mean curvature. This is done essentially by the characteristicpolynomial of the shape operator.

For an (n× n)-matrix A, we consider the polynomial

PA(t) = det(A+ tI) = antn + ...+ a0.

Then if Q is an invertible matrix, we have

39

PQAQ−1(t) = det(QAQ−1 + tI) = det(QAQ−1 + tQQ−1)

= det(Q) det(Q−1) det(A+ tI) = det(A+ tI) = PA(t)

so that the polynomial is independent of the choice of basis.

With a little thought, see page 54 of Lancaster [4], one realises that the coefficientak is given by

ak =∑

1≤i1<...<ik≤n

[A]i1...ik ,

where [A]i1...ik denotes the principal minor determinant of A omitting the rows andcolumns i1 . . . ik. And using the notation as described in Section 1.3 we may write

ak =∑

J∈Ik(n)

[A]J

i.e. the sum of the determinants of the principal (n−k)×(n−k) submatrices. Sincethe polynomial PA is independent if the representation of A, so are the coefficientsak. The number of terms in each sum is

(nk

). Thus we can produce an average.

Definition 4.23. For a matrix A we denote the average of the determinants of theprincipal submatrices of dimension k with Hk(A) and we have(

n

k

)Hk(A) =

∑J∈In−k(n)

[A]J .

It is clear from the characteristic polynomial, that this definition is equivalent toDefinition 4.18 when the matrix A is the local representation of the shape operatorS.

Now we can write the two previous integral identities derived in Examples 4.20and 4.22 as. ∫

M

H0(S)dVolM =

∫M

H1(S)〈N, p〉dVolM

and ∫M

Hm−1(S)dVolM =

∫M

Hm(S)〈N, p〉dVolM .

These are special cases of Theorem 4.19. If open for some combinatorial chaos, wemay prove the whole theorem by similar computation.

Proof. (Theorem 4.19) The proof will follow a similar method as the Examples 4.20and 4.22. For a given k we define a suitable (m−1)-form ω and show that it satisfies

dωp = C(Hk+1(Sp)〈Np, p〉 −Hk(Sp)

)for a non zero constant C. Then by Stokes’ Theorem on manifolds we get∫

M

dω =

∫∂M

ω =

∫p

ω = 0.

It therefore follows that

40

∫M

Hk(S)dVolM =

∫M

Hk+1(S)〈N, p〉dVolM

and the result will be established. Example 4.20 is a proof of Theorem 4.18 for thecase k = 0 and dim(M) = 2. A similar form suffices for any dimension m, that is ωgiven for X1, . . . , Xm commuting vector fields by

ωp =m∑i=1

αp(X1, . . . , Xi, . . . , Xm)dx1 . . . dxi . . . dxm

whereαp(X1, . . . , Xi, . . . , Xm) = det(p,Np, X1, . . . , Xi, . . . , Xm).

Example 4.22 is a proof for the case k = m− 1. then ωp is given by the coefficients

αp(X1, . . . , Xi, . . . , Xm) = det(p,Np, Sp(X1), . . . , Sp(Xi), . . . , Sp(Xm).

In both these examples the resulting derivative dω can be separated into threegroups of terms. Two of them form the terms of the final equality, and the lastgroup consists of terms that cancel pairwise.

For any k = 1, . . . ,m− 1, we may define the form ω′ by

ω′ =m∑i=1

α′(X1, . . . , Xi, . . . , Xm)dx1 . . . dxi . . . dxm

where

α′p(X1, . . . , Xi, . . . , Xm) = det(p,Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm).

This is a form similar to the two previous ones, but with the shape operator appliedto k of the vectors, not none or all. This is a good start, but we are not there yet.if k is not 0 or m, there is room for choice in defining α′. What we will need is thesum of all these possible choices.

For a given k, let ω be the (m− 1)-form

ω =m∑i=1

α(X1, . . . , Xi, . . . , Xm)dx1 . . . dxi . . . dxm

where α is defined by

αp(X1, . . . , Xi, . . . , Xm) =∑

J∈Ik(m\i)

det(p,Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

where J = (j1, . . . , jk). This is a sum of(m−1k

)terms. Then

dω = −m∑i=1

(−1)iXi

(∑J∈Ik(m\i)

det(p,Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm))dVolp

is a sum of m ·(m−1k

)terms. Now for each i we get the factors such that by the

product rule we get

Xi

( ∑J∈Ik(m\i)

det(p,Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

)41

=∑

J∈Ik(m\i)

det(Xi, Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

+∑

J∈Ik(m\i)

det(p, Sp(Xi), X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

+∑l 6=i

( ∑J∈Ik(m\i,l)

det(p,Np, X1, . . . , Sp(Xj1), . . . ,∇XiXl, . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

+∑

J∈Ik−1(m\i,l)

det(p,Np, X1, . . . , Sp(Xj1), . . . , (∇XiS(Xl))p, . . . , Xi, . . . , Sp(Xjk−1), . . . , Xm)

).

As before, in Example 4.20 and 4.22, each term of the last double sum, has acomplementary term in the l:th term, that together forms a pair that cancels outby Lemma 4.11. We focus our attention on the remaining two sums for each i. Ifwe sort the Xi and Sp(Xi) into the i-th place sum and over all i, we obtain

−m∑i=1

∑J∈Ik(m\i)

det(Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

+m∑i=1

∑J∈Ik(m\i)

det(p,X1, . . . , Sp(Xj1), . . . , Sp(Xi), . . . , Sp(Xjk), . . . , Xm).

These are sums of m ·(m−1k

)terms. The question is, how many terms in each sum

are equal?

For the first sum, each specific term is a determinant, with the normal as firstentry, and the shape operator applied to k of the local basis vectors of M;

det(Np, X1, . . . , Sp(Xj1), . . . , Sp(Xjk), . . . , Xn).

A specific term of this sum will be obtained when the first term in the determinantis differentiated (Xi(p) = Xi), and equal terms are obtained exactly once for eachi /∈ j1, . . . , jk. So that Xi appears in the term without the shape operator appliedto it. That is (m− k) times, and we can write

m∑i=1

∑J∈Ik(m\i)

det(Np, X1, . . . , Sp(Xj1), . . . , Xi, . . . , Sp(Xjk), . . . , Xm)

= (m− k)∑J∈Im−k(m)

[Sp]J

= (m− k)

(m

m− k

)Hk(Sp)

In the second sum, the specific term is the determinant, with the point p as thefirst entry, and the shape operator applied to a total of (k + 1) of the local basisvectors of M, i.e.

det(p,X1, . . . , Sp(Xj1), . . . , Sp(Xjk+1), . . . , Xm).

These are obtained when differentiating the normal vector field

(∇XiN) = S(Xi)

42

and equal terms are obtained exactly once for each i ∈ j1, . . . , jk+1 where theshape operator is applied to the vector fields Xj, and therefore will be obtainedexactly (k + 1) times. This yields

m∑i=1

∑J∈Ik(m\i)

det(p,X1, . . . , Sp(Xj1), . . . , Sp(Xi), . . . , Sp(Xjk), . . . , Xm)

= (k + 1)∑

J∈Im−(k+1)(m)

[Sp]J〈Np, p〉

= (k + 1)

(m

k + 1

)Hk+1(Sp)〈Np, p〉

It remains to convince ourselves that these partially binomial factors are equal.

(m− k)

(m

m− k

)=

(m− k)m!

k!(m− k)!=

m(m− 1)!

k!(m− 1− k)!= m

(m− 1

k

)and

(k + 1)

(m

k + 1

)=

(k + 1)m!

(k + 1)!(m− (k + 1))!=

m(m− 1)!

k!(m− 1− k)!= m

(m− 1

k

).

Finally

dωp = m

(m− 1

k

)(−Hk(Sp) +Hk+1(Sp)〈Np, p〉)dVolp

And by the General Theorem of Stokes and the fact that ∂M = 0 we get∫M

dω =

∫∂M

ω = 0.

We have now established the relation∫M

Hk(S)dVolM =

∫M

Hk+1(S)〈N, p〉dVolM

for k = 0, . . . ,m− 1.

43

44

Bibliography

[1] M. Spivak, Calculus on manifolds. A modern approach to classical theorems ofadvanced calculus, W. A. Benjamin Inc. (1965).

[2] H. Flanders, Differential Forms with Applications to the Physical Sciences,Dover Publications (1989).

[3] M.P. doCarmo, Differential Forms and Applications, Springer (1994).

[4] P. Lancaster, Theory of Matrices, Academic Press (1969).

[5] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics 171,Springer (1998).

[6] S. Gudmundsson, An Introduction to Gaussian Geometry, Lecture Notes inMathematics, Lund University (2018).

[7] S. Gudmundsson, An Introduction to Riemannian Geometry, Lecture Notes inMathematics, Lund University (2018).

[8] S. Gudmundsson, handwritten lecture notes on isometric ovaloids, Universityof Bonn (1985).

[9] J. Munkres, Topology, Pearson (2004).

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45

46

Bachelor’s Theses in Mathematical Sciences 2019:K10ISSN 1654-6229

LUNFMA-4088-2019Mathematics

Centre for Mathematical SciencesLund University

Box 118, SE-221 00 Lund, Swedenhttp://www.maths.lth.se/