NATURAL METRICS

ON

TANGENT BUNDLES

ELIAS KAPPOS

Master’s thesis

2001:E10

Centre for Mathematical Sciences

Mathematics

CE

NT

RU

MSC

IEN

TIA

RU

MM

AT

HE

MA

TIC

AR

UM

Abstract

In this Master0s dissertation we study the geometry of tangent bundles TMof Riemannian manifolds (M ; g). We calculate their Lie bracket, introduce anatural class of Riemannian metrics g on the tangent bundle TM transform-ing the natural projection from the tangent bundle TM onto (M ; g) into aRiemannian submersion. As examples of natural metrics we discuss in detailthe Sasaki and the Cheeger-Gromoll metrics. We calculate their Levi-Civitaconnections and various curvatures. This leads to some interesting connec-tions between the geometry of the Riemannian manifold (M ; g) and its tan-gent bundle TM equipped with these two natural metrics.

Throughout Chapters 2, 3 and 4 of this work it has been my firm intentionto give reference to the stated results and credit the work of others. The results andproofs not marked with a number in brackets [X ] are the fruits of my efforts.

Acknowledgements

I am grateful to all those who helped me improve this work by usefulcomments and suggestions, in particular my supervisor Dr. Sigmundur Gud-mundsson to whom I am also grateful for his patience and encouragment, Prof.Masami Sekizawa for discussions on Chapter 4 and Martin Svensson.

I would also like to thank the Mathematical Faculty at the University ofGottingen for giving me the opportunity to study mathematics in Lund on aone year Erasmus scholarship at the Centre for Mathematical Sciences, LundUniversity.

Elias Kappos

Contents

Introduction 3

Chapter 1. The Tangent Bundle 51. Differentiable Manifolds 52. Tangent Spaces 63. The Tangent Bundle 84. The Lie Bracket 115. Riemannian Metrics 126. The Levi-Civita Connection 137. The Riemann Curvature Tensor 15

Chapter 2. Structures on Tangent Bundles 191. The Vertical and Horizontal Lifts 192. The Lifts in Local Coordinates 223. The Lie Bracket 244. The Natural Almost Complex Structure 265. Natural Metrics 27

Chapter 3. The Sasaki Metric 331. The Levi-Civita Connection 332. The Curvature Tensor 353. Geometric Consequences 38

Chapter 4. The Cheeger-Gromoll Metric 431. The Levi-Civita Connection 432. The Curvature Tensor 463. Geometric Consequences 52

Bibliography 63

1

Introduction

The main purpose of this Master0s project is to write a survey on the geom-etry of tangent bundles. This is an area which has been developed over severaldecades and the authors have used a variety of approaches and very differentnotation. It is our main aim to write a unified presentation of some of the bestknown results in this field.

The research in this area of differential geometry began with the funda-mental paper [16] of Sasaki published in 1958. He uses the metric g on aRiemannian manifold (M ; g) and its Levi-Civita connection r to construct ametric g on the tangent bundle TM of M . Today this metric is a standardnotion in Differential Geometry called the Sasaki metric.

In his famous article [6] from 1962, Dombrowski calculates the Lie bracket[; ] on the tangent bundle TM and discusses its natural almost complex struc-ture J , constructed by Nagano in [14].

In 1971 Kowalski published [10] where he uses the Lie bracket on TM to

get explicit formulae for the Levi-Civita connection r of the tangent bundle(TM ; g) equipped with the Sasaki metric. Then he calculates the correspond-

ing Riemann curvature tensor R of (TM ; g).

In 1981 Aso [1] and in 1988 Musso and Tricerri [13] use Kowalski0s cal-culations of the curvature tensor to draw some very interesting conclusions onthe connection between the geometry of (M ; g) and that of (TM ; g).

As early as 1972 Cheeger and Gromoll suggested a way of constructinga new natural Riemannian metric ~g on the tangent bundle TM of (M ; g).This heavily depends on Sasaki0s idea of splitting the tangent bundle TTM ofTM into a horizontal and a vertical part, using the Levi-Civita connection rof (M ; g). The first explicit expression of the Cheeger-Gromoll metric ~g wasgiven in the paper [13] of Musso and Tricerri published in 1988. Sekizawa uses

this in [18] to calculate both, the Levi-Civita connection ~r and the curvaturetensor ~R of (TM ; ~g). With this in hand he was able to give an interestingconnection between the geometry of (M ; g) and that of (TM ; ~g).

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In Chapter 1 we lay the basis for this work. We introduce the conceptof a differentiable manifold M and its tangent bundle TM , the Levi-Civitaconnection r and the Riemann curvature tensor R. We discuss widely knownfacts, which are only proven in certain exceptional cases. They can be foundin any book on Riemannian geometry, see for example [5] or [8].

In Chapter 2 we equip the tangent bundle with additional structures suchas horizontal and vertical lifts of vector fields on M and an almost complexstructure. We calculate the Lie bracket, define a class of natural metrics g on

TM and obtain formulae for its Levi-Civita connection r. In this chapter wemainly follow the work of Dombrowski [6].

Chapter 3 is devoted to the Sasaki metric as an example of a natural metric.We calculate its Levi-Civita connection and its Riemann curvature tensor andconclude with some geometric consequences. In this chapter we follow thework of Kowalski [10], Aso [1] and Musso-Tricerri [13].

Chapter 4 is devoted to the Cheeger-Gromoll metric. We calculate theLevi-Civita connection, the Riemann curvature tensor and obtain some geo-metric conclusions. Here we mainly follow the work of Musso-Tricerri [13]and Sekizawa [18], but without using the methods of moving frames, as in[13]. In the last section of this chapter we have found and corrected somemistakes in [18].

Throughout this work we have used Riemannian manifolds of constantsectional curvature as an example to illustrate the main results.

For later development in this field we recommend the reading of [11], [4]and [12].

4

CHAPTER 1

The Tangent Bundle

In this chapter we introduce the main object under investigation in thiswork, namely the tangent bundle TM of a smooth manifold M . This is itselfa smooth manifold of double the dimension of that of M . In order to set upthe notation being used we give a very brief introduction to smooth manifoldsfollowing the presentation of [8]. The missing proofs are standard and can befound in most books on Riemannian geometry, see for example [5], [7] or [9].

1. Differentiable Manifolds

Definition 1.1. Let M be a topological Hausdorff space with a countablebasis. M is called a topological manifold, if there exists an m 2 N and for everypoint p 2 M an open neighborhood Up of p such that Up is homeomorphic tosome open subset Vp � Rm . The integer m is called the dimension of M . Wewrite Mm to denote that M has dimension m.

Definition 1.2. Let Mm be a topological manifold, U an open and con-nected subset of M and f : U ! Rm a continuous map homeomorphic ontoits image f(U ). Then (U ;f) is called a local coordinate on M . A collectionA = f(Ua;fa) j a 2 Ig of local coordinates on M is called a C r-atlas if

i) M = Sa Ua, andii) the corresponding transition mapsfb Æ f�1a jfa(Ua\Ub ): fa(Ua \ Ub ) ! Rm

are C r for all a; b 2 I .

If A is a C r-atlas on M then a local coordinate (U ;f) on M is said to be

compatible with A if A [ f(U ;f)g is a C r-atlas. A C r-atlas A is maximal if itcontains all local coordinates compatible with it. It is also called a C r-structure

on M and the pair (M ; A) is called a differentiable C r-manifold. By smooth wemean C1 defined by C1 = T1

k=1C k.

We now are able to discuss maps between two differentiable manifolds. Weare mainly interested in differentiable maps, so we will now define:

Definition 1.3. Let (Mm; A) and (N n; B) be two C r-manifolds. A mapy : Mm ! N n is said to be a C r-map if for all local coordinates (U ;f) 2 A5

and (V ; q) 2 B the mapsq Æ y Æ f�1 jf(U\y�1(V )): f(U \ y�1(V )) ! Rn

are of class C r . If y maps to R it is called a C r-function on M .

In what follows we shall assume that all manifolds and maps are smooth,i.e. in the C1-category.

Proposition 1.4. Let f : (M ; A) ! (N ; B) and y : (N ; B) ! (L; C) be

two C1-maps, then the composition y Æ f : (M ; A) ! (L; C) is also a C1-map.

Definition 1.5. Two C r-manifolds (M ; A) and (N ; B) are said to be dif-

feomorphic if there exists a bijective C1-map f : (M ; A) ! (N ; B) such that

its inverse also is C1. The map f is called a diffeomorphism between (M ; A)

and (N ; B). If N is equal to M then the two C1-structures A and B are said

to be different, if the identity map idM : (M ; A) ! (M ; B) is not a diffeomor-phism.

2. Tangent Spaces

In this section we introduce the tangent space TpM of M at a point p. LetMm be a manifold and p 2 M a point of M and e(p) be the set of all smoothfunctions on an open neighborhood around p, i.ee(p) = ff : Uf ! R j Uf � M open and containing pg:Now we define an equivalence relation on e(p) by: f � g if and only if thereexists an open neighborhood V � Uf \ Ug such that f jV= g jV . By e(p)

we denote the set of equivalence classes of elements in e(p). The elements f ofe(p) are called function germs at p. We can turn e(p) into an R-algebra by thefollowing operations + and �

i) f + g = f + g ,

ii) l � f = l � f ,

iii) f � g = f � g

for all f ; g 2 e(p) and l 2 R. From now on we will denote the elements ofe(p) by f instead of f .

Definition 1.6. A tangent vector Xp at p 2 M is a map Xp : e(p) ! R suchthat

i) Xp(l � f + m � g) = l � Xp(f ) + m � Xp(g),ii) Xp(f � g) = g(p) � Xp(f ) + f (p) � Xp(g)

for all l; m 2 R and f ; g 2 e(p). The set of all tangent vectors Xp at p 2 M isdenoted by TpM and is called the tangent space of M at p.

The tangent space TpM is turned into a real vector space by defining theoperations + and � by

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i) (Xp + Yp)(f ) = Xp(f ) + Yp(f ),ii) (l � Xp)(f ) = l � Xp(f )

for all l 2 R and Xp; Yp 2 TpM .For M = Rm we denote em by e(0) as the set of function germs at 0 2 R.

It is well known from calculus, that for v 2 Rm and f 2 em the directionalderivative of f at 0 in the direction of v is given by�vf = lim

t!0

f (tv)� f (0)

t:

For v = (v1; : : : ; vm) we get �vf =Pmi=1

vi�f�xi

(0) and we know that

i) �v(l � f + m � g) = l � �vf + m � �vg,ii) �v(f � g) = g(0) � �vf + f (0) � �vg,

iii) �l�v+m�wf = l � �vf + m � �wg

for all l; m 2 R, v;w 2 Rm and f ; g 2 em. Hence �v is an element of T0Rm .

Theorem 1.7. The map F : Rm ! T0Rm given by v 7! �v is a vector spaceisomorphism.

Corollary 1.8. Let fek j k = 1; : : : ;mg be a basis for Rm . Thenf�ekj k = 1; : : : ;mg

is a basis for the tangent space T0Rm .

Definition 1.9. Let f : M ! N be a map between two manifolds. For apoint p 2 M we define the map dfp : TpM ! Tf(p)N by

(dfp)(Xp)(f ) = Xp(f Æ f)

for all Xp 2 TpM and f 2 e(f(p)). The map dfp is called the differential of fat p 2 M .

Proposition 1.10. Let f : M ! ~M and y : ~M ! N be two maps betweensmooth manifolds, then

i) the map dfp : TpM ! Tf(p)~M is linear,

ii) if idM is the identity map, then d (idM )p = idTpM ,iii) d (y Æ f)p = dyf(p) Æ dfp

for all p 2 M. The equation iii) is called the Chain rule.

PROOF. The first two points follow directly from the definition, so weonly have to prove the chain rule. If Xp 2 TpM and f 2 e(y Æ f(p)), then

(dyf(p) Æ dfp(Xp))(f ) = (dfp(Xp))(f Æ y)= Xp(f Æ y Æ f)= d (y Æ f)p(Xp)(f ): �7

Corollary 1.11. Let f : M ! N be a diffeomorphism with inverse y =f�1 : N ! M. Then the differential dfp : TpM ! Tf(p)N at p is bijective and(dfp)

�1 = dyf(p).

As a direct consequence of Corollary 1.8 and Corollary 1.11 we obtain

Corollary 1.12. Let p be a point of a m-dimensional manifold M. Then thetangent space TpM at p is an m-dimensional real vector space.

Definition 1.13. Let Mm be a manifold, (U ; x) be a local coordinate onM and fek j k = 1; : : : ;mg be the standard basis for Rm . For p 2 M define( ��xk

)p 2 TpM by

(��xk

)p : f 7! �f�xk

(p) = �ek(f Æ x�1)(x(p)):

Proposition 1.14. The set f( ��xk)p j k = 1; : : : ;mg is a basis for TpM for

all p 2 U .

PROOF. Because M is smooth, it follows that the inverse of x is smoothand therefore the differential of the inverse satisfies

(dx�1x(p)

(�ek)(f ) = �ek

(f Æ x�1)(x(p)) = ((��xk

)p)(f )

for all f 2 e(p). �The tangent space TpM may be viewed in an alternative way. For this we

use the set C (p) of all equivalence classes of locally defined C1-curves passingthrough the point p 2 M . It is possible to identify TpM with C (p) being theset of all tangents to curves going through the point p. Then a vector v 2 TpMcan be discribed by

v(f ) = d

dt(f Æ g(t))jt = 0;

with f : U � M ! R a function defined on U containing p and g : I ! U

an arbitrary curve with g(0) = p and�g (0) = v.

3. The Tangent Bundle

In this section we introduce the notion of a smooth vector bundle over amanifold. The tangent bundle turns out to be an important example and themain object in this work.

Definition 1.15. Let E and M be topological manifolds andp : E ! M be a continuous surjective map. If

i) for each p 2 M the fiber Ep = p�1(p) is an n-dimensional vector spaceand

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ii) for each p 2 M there exists a bundle chart (p�1(U );y) consisting ofthe pre-image of p of an open neighborhood U of p and a homeo-morphism y : p�1(U ) ! U � Rn such that for all q 2 U the mapyq = y jEq : Eq ! fqg � Rn is a vector space isomorphism,

then the triple (E ;M ; p) is called an n-dimensional topological vector bundleover M. It is said to be trivial if there exists a global bundle chart y : E !M � Rn .

Definition 1.16. Let (E ;M ; p) be a topological vector bundle. A contin-uous map s : M ! E is called a section of the bundle if p Æ s(p) = p for eachp 2 M .

Definition 1.17. A collectionB = f(p�1(Ua);ya) j a 2 Igof bundle charts is called a bundle atlas for (E ;M ; p) if M = Sa Ua. For eachpair (a; b) there exists a function Aa;b : Ua \ Ub ! GL(Rn ), into the generallinear group GL(Rn ) of Rn , such that the corresponding continuous mapya Æ y�1b j(Ua\Ub )�Rn : (Ua \ Ub )� Rn ! (Ua \ Ub )� Rn

is given by(p; v) 7! (p; (Aa;b )(v)):

The elements of fAa;b j a; b 2 Ig are called the transition maps of the bundleatlas B.

Remark 1.18. Since all the maps which we are using are smooth we calla topological vector bundle smooth, if B is maximal. A smooth section of(E ;M ; p) is called a vector field and we denote the set of all vector fields of(E ;M ; p) by C1(E).

Definition 1.19. By the following operations we make C1(E) into aC1(M ) = C1(M ;R) module

i) (v + w)p = vp + wp,ii) (f � v)p = f (p) � vp

for all v;w 2 C1(E) and f 2 C1(M ) = C1(M ;R). In particular, C1(E) isa vector space over the real numbers.

Definition 1.20. Let M be a manifold and (E ;M ; p) be an n-dimensionalvector bundle over M . A set F = fv1; : : : ; vng of vector fields

v1; : : : ; vn : U � M ! E

is called a local frame for E over U if for each p 2 U the set f(v1)p; : : : ; (vn)pgis a basis for the vector space Ep.

We can now define the tangent bundle TM on M . It can be thought of asthe object you get by gluing the tangent space TpM at p on M for all p 2 M .This way we obtain a 2m-dimensional topological vector bundle.

9

Definition 1.21. Let (M ; A) be a manifold. The tangent bundle TM ofM is given by

TM = f(p; u) j p 2 M ; u 2 TpMg:The bundle map p : TM ! M with p : (p; u) 7! p is called the naturalprojection of TM .

Theorem 1.22. Let M be a topological manifold of dimension m with a C1-atlas A. Then the tangent bundle TM is a topological manifold of dimension 2mand A induces a C1-atlas A� on TM.

PROOF. For every local coordinate x : U ! Rm in A we define x� :p�1(U ) ! Rm � Rm by

x� : (p; mXk=1

uk��xkjt) 7! (x(p); (u1; : : : ; um)):

Then the collectionf(x�)�1(W ) � TM j (U ; x) 2 A and W � x(U )� Rm opengis a basis for a topology TTM on TM and (p�1(U ); x�) is a local coordinate onthe 2m-dimensional topological manifold (TM ; TTM ).

If (U ; x) and (V ; y) are two local coordinates in A such that p 2 U \ V ,then the transition map

(y�) Æ (x�)�1 : x�(p�1(U \ V )) ! Rm � Rm

is given by

(p; u) 7! y Æ x�1(x); mX

k=1

�y1�xk

(x�1(p))uk; : : : ; mXk=1

�ym�xk

(x�1(p))uk

!We are assuming that y Æ x�1 is smooth, hence (y�) Æ (x�)�1 is smooth and

therefore A� = f(p�1(U ); x�) j (U ;f) 2 Ag is a C1-atlas on TM and

(TM ; A�) is a smooth manifold. �Remark 1.23. For each point p 2 M the fiber p�1(p) of p is the tangent

space TpM of M at p and hence an m-dimensional vector space. For a local

coordinate x : U ! Rm 2 A we define x : p�1(U ) ! U � Rm by

x : (p; mXk=1

uk��xk

jp) 7! (p; (u1; : : : ; um)):The restriction xp = xjTpM : TpM ! fpg � Rm to TpM is given by

xp :

mXk=1

uk��xkjp 7! (u1; : : : ; um);10

which obviously is a vector space isomorphism. Hence the x : p�1(U ) !U � Rm is a bundle chart. This implies thatB = f(p�1(U )); x) j (U ; x) 2 Agis a bundle atlas transforming (TM ;M ; p) into an m-dimensional topologicalvector bundle. This implies that the vector bundle (TM ;M ; p) together with

the maximal bundle atlas B induced by B is a smooth vector bundle.

4. The Lie Bracket

In this section we introduce the Lie bracket on a differentiable manifold(M ;A).

Definition 1.24. Let M be a manifold and X ; Y 2 C1(TM ) be vectorfields on M . Then the Lie bracket [X ; Y ]p of X and Y at p 2 M is defined by

[X ; Y ]p(f ) = Xp(Y (f ))� Yp(X (f )) 2 R;where f 2 C1(M ).

Lemma 1.25. Let M be a smooth manifold, X ; Y 2 C1(TM ) be two vectorfields on M, f ; g 2 C1(M ) and l; m 2 R. Then

i) [X ; Y ]p(lf + mg) = l[X ; Y ]p(f ) + m[X ; Y ]p(g),ii) [X ; Y ]p(f � g) = f (p)[X ; Y ]p(g) + g(p)[X ; Y ]p(f ).

Proposition 1.26. Let M be a manifold and X ; Y 2 C1(TM ) be vectorfields on M. Then

i) [X ; Y ]p is an element of TpM for all p 2 M,ii) the section [X ; Y ] : p 7! [X ; Y ]p is smooth.

Theorem 1.27. Let M be a smooth manifold. Then the vector space C1(TM )of smooth vector fields on M equipped with the Lie bracket is a Lie algebra over thereal numbers i.e.

i) [lX + mY ;Z ] = l[X ;Z ] + m[Y ;Z ],ii) [X ; Y ] = �[Y ;X ],

iii) [X ; [Y ;Z ]] + [Z ; [X ; Y ]] + [Y ; [Z ;X ]] = 0,

for all X ; Y ;Z 2 C1(TM ) and l; m 2 R. The equation iii) is called the Jacobiidentity for C1(TM ).

Lemma 1.28. Let M be a smooth manifold. Then

i) [X ; f � Y ] = X (f ) � Y + f � [X ; Y ];ii) [f � X ; Y ] = f � [X ; Y ]� Y (f ) � X

for all X ; Y 2 C1(TM ) and f 2 C1(M ).

Definition 1.29. Let M be a smooth manifold. Two vector fields X ; Y 2C1(TM ) are said to commute if [X ; Y ] = 0.

Lemma 1.30. Let f : M ! N be a smooth bijective map between twomanifolds. If X ; Y 2 C1(TM ) are vector fields on M, then

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i) df(X ) 2 C1(TN ),ii) the map df : C1(TM ) ! C1(TN ) is a Lie algebra homomorphism

i.e. [df(X ); df(Y )] = df([X ; Y ]).In this case we say that X ; Y and df(X ); df(Y ) are f-related.

PROOF. The first statement follows directly from the definition of the dif-ferential. Let f : N ! R be smooth, then

[df(X ); df(Y )](f ) = df(X )(df(Y (f ))� df(Y )(df(X (f ))= X (df(Y )(f ) Æ f)� Y (df(X )(f ) Æ f)= X (Y (f Æ f))� Y (X (f Æ f))= [X ; Y ](f Æ f) = df([X ; Y ])(f ): �5. Riemannian Metrics

In this section we will introduce the notion of a Riemannian manifold. Itwill be defined as a manifold with a certain symmetric (2; 0)-tensor.

Definition 1.31. Let M be a smooth manifold, C1(M ) the commutativering of smooth functions on M and C1(TM ) the module over C1(M ) ofsmooth vector fields on M . Put C1

0 (TM ) = C1(M ) and for k 2 N+ let

C1k (TM ) = C1(TM ) � � � C1(TM );

be the k-fold tensor product of C1(TM ). A tensor field B on M of type (r; s)is a map B : C1

r (TM ) ! C1s (TM ) satisfying

B(X1 � � � Xl�1 (f � Xl + g � Y ) Xl+1 � � � Xr)= f � B(X1 � � � Xr)+ g � B(X1 � � � Xl�1 Y Xl+1 � � � Xr)

for all X1; : : : ;Xr; Y 2 C1(TM ), f ; g 2 C1(M ) and l = 1; : : : ; r.

¿From now on we use the notation B(X1; : : : ;Xr) for B(X1 � � � Xr).

Proposition 1.32. Let B be a tensor field of type (r; s), p 2 M a point of themanifold m and X1; : : : ;Xr, Y1; : : : ; Yr be smooth vector fields on M such that(Xk)p = (Yk)p for each k = 1; : : : ; r. Then

B(X1; : : : ;Xr)(p) = B(Y1; : : : ; Yr)(p):By Bp we denote the restriction Bp = B jLr

l=1TpM of B to the r-fold tensor

product of TpM given by

Bp : ((X1)p; : : : ; (Xr)p) 7! B(X1; : : : ;Xr)(p):12

Definition 1.33. The tensor field B is said to be smooth if for all X1; : : : ;Xr2 C1(TM ) the map B(X1; : : : ;Xr) : M ! C1s (TM ) with

B(X1; : : : ;Xr) : p 7! Bp((X1)p; : : : ; (Xr)p)

is smooth.

Definition 1.34. Let (M ;A) be a smooth manifold. A smooth tensorfield g : C1

2 (TM ) ! C10 (M ) is called a Riemannian metric on M if for each

p 2 M the restriction gp : TpM � TpM ! R with

gp : (Xp; Yp) 7! g(X ; Y )(p)

is an inner product on TpM . The pair (M ; g) is called a Riemannian manifold.

Theorem 1.35. Let (Mm;A) be a smooth manifold. Then there exists aRiemannian metric g on M.

Definition 1.36. Let (M ; g) and (N ; h) be Riemannian manifolds. A mapf : M ! N is said to be conformal if there exists a positive function l : M !R+ such that l(p)gp(Xp; Yp) = hf(p)(dfp(Xp); dfp(Xp));for all X ; Y 2 C1(TM ) and p 2 M . The function l is called the conformalfactor of f. A conformal map with l � 1 is said to be isometric. An isometricdiffeomorphism is called an isometry.

6. The Levi-Civita Connection

Later on we will discuss geometric properties of Riemannian manifolds. Tobe able to do this we must first explain how we can differentiate a vector fieldin the direction of another vector field. This will be done in this section, wherewe define the Levi-Civita connectionr on M , which is the unique metric andtorsion-free connection on the manifold M .

Definition 1.37. Let (E ;M ; p) be a smooth vector bundle over M . A con-

nection�r on (E ;M ; p) is a map

�r: C1(TM )� C1(E) ! C1(E) satisfying

i)�rX (l � v + m � w) = l� �rX v + m� �rX w

ii)�rX (f � v) = X (f ) � v + f � �rX v

iii)�rf �X+g�Y v = f � �rX v + g� �rY v

for all l; m 2 R, X ; Y 2 C1(TM ), v;w 2 C1(E) and f ; g 2 C1(M ), such

that the map (�rX v) : M ! E with (

�rX v) : p 7! (�rX v)p is smooth. A

section v 2 C1(E) is said to be parallel with respect to�r if

�rX v = 0 for allX 2 C1(TM ).

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Definition 1.38. Let M be a smooth manifold and�r be a connection on

the tangent bundle (TM ;M ; p). Then the torsion T : C12 (TM ) ! C1(TM )

of�r is defined by

T (X ; Y ) = �rX Y� �rY X � [X ; Y ]:The connection

�r is said to be torsion-free if the corresponding torsion Tvanishes i.e.

T (X ; Y ) = 0

for all X ; Y 2 C1(TM ).

Definition 1.39. Let (M ; g) be a Riemannian manifold. Then�r is said

to be compatible with g (or metric) if

X (g(Y ;Z )) = g(�rX Y ;Z ) + g(Y ; �rX Z )

for all X ; Y ;Z 2 C1(TM ).

Theorem 1.40. Let (M ; g) be a Riemannian manifold and let the map r :C1(TM )� C1(TM ) ! C1(TM ) be given by the so called Koszul formula

g(rXY ;Z ) = 1

2(X (g(Y ;Z )) + Y (g(Z ;X ))� Z (g(X ; Y ))+g(Z ; [X ; Y ]) + g(Y ; [Z ;X ])� g(X ; [Y ;Z ])):

Then r is a connection on the tangent bundle (TM ;M ; p).

PROOF. This theorem follows directly by the fact that g is a tensor fieldand by Definition 1.6, Theorem 2.11 and Lemma 1.28. �

Definition 1.41. The connection r on (M ; g) defined in the Theorem1.40 is called the Levi-Civita connection of g.

The next result is called The Fundamental Theorem of Riemannian Ge-ometry.

Theorem 1.42. Let (M ; g) be a Riemannian manifold. Then the Levi-Civitaconnection r is the unique metric and torsion-free connection on (TM ;M ; p).

Definition 1.43. Let (M ; g;r) be a Riemannian manifold with its Levi-Civita connection. Further let (U ; x) be a local coordinate on M and defineXi = ��xi

2 C1(TU ). Then fX1; : : : ;Xmg is a local frame of TM on U . For

(U ; x) the Christoffel symbols G kij : U ! R of r with respect to (U ; x) are

given bymX

k=1

G kij Xk = rXi

Xj:14

7. The Riemann Curvature Tensor

In this section we define the curvature tensor R, which is an important toolfor understanding the geometry of the manifold (M ; g).

Definition 1.44. Let (M ; g;r) be a Riemannian manifold with its Levi-Civita connection. For i 2 f0; 1g and a tensor field A : C1

r (TM ) !C1

i (TM ) we define its covariant derivative rA : C1r+1(TM ) ! C1

i (TM ),by rA :(X ;X1; : : : ;Xr) 7! (rXA)(X1; : : : ;Xr) =

X (A(X1; : : : ;Xr))� rXi=1

A(X1; : : : ;Xi�1;rXXi;Xi+1; : : : ;Xr)

A tensor field A of type (r,0) or (r,1) is said to be parallel if rA � 0.

A vector field Z 2 C1(TM ) defines a smooth tensor field Z : C1(TM ) !C1(TM ) given by Z : X 7! rXZ . For two vector fields X ; Y 2 C1(TM ) we

define the second covariant derivative r2X ;Y : C1(TM ) ! C1(TM ) byr2

X ;Y : Z 7! (rXZ )(Y ):Hence r2

X ;Y Z = rX(Z (Y ))� Z (rXY ) = rXrYZ �rrXYZ :Definition 1.45. Let (M ; g;r) be a Riemannian manifold with the Levi-

Civita connection. Let R : C13 (TM ) ! C1

1 (TM ) be twice the skrew-sym-metric part of the second covariant derivative r2 i.e.

R(X ; Y )Z = r2X ;Y Z �r2

Y ;X Z = rXrYZ �rYrXZ �r[X ; Y ]Z :Then R is called the Riemann curvature tensor of (M ; g).

The next result shows that the curvature tensor has many nice symmetricproperties.

Proposition 1.46. Let (M ; g) a Riemannian manifold. Then following iden-tities hold

i) R(X ; Y )Z = �R(Y ;X )Zii) g(R(X ; Y )Z ;W ) = �g(R(X ; Y )W ;Z )

iii) g(R(X ; Y )Z ;W ) + g(R(Z ;X )Y ;W ) + g(R(Y ;Z )X ;W ) = 0iv) g(R(X ; Y )Z ;W ) = g(R(Z ;W )X ; Y )

The equation given in iii) is called the 1st Bianchi identity.

For all p 2 M let G2(TpM ) denote the set of all 2�dimensional subspacesof TpM .

15

Lemma 1.47. Let X ; Y ;Z ;W 2 TpM be vectors in the tangent space at thepoint p such that spanRfX ; Y g = spanRfZ ;W g. Then

g(X ; Y )Y ;X )jX j2jY j2 � g(X ; Y )2= g(Z ;W )W ;Z )jZ j2jW j2 � g(Z ;W )2

Definition 1.48. For a point p 2 M the function Kp : G2(TpM ) ! Rgiven by

Kp : spanRfX ; Y g 7! g(R(X ; Y )Y ;X )jX j2jY j2 � g(X ; Y )2

is called the sectional curvature of the 2-plane spanned by X and Y at p. TheRiemannian manifold (M ; g) is said to be of constant sectional curvature if thefunction Kp is constant for all p 2 M and all X ; Y 2 TpM . The Riemannianmanifold (M ; g) is said to be flat if its constant sectional curvature is zero.

In the case of constant sectional curvature the Riemann curvature tensorhas a rather simple form.

Lemma 1.49. Let (M ; g) be a Riemannian manifold of constant sectionalcurvature k. Then the Riemann curvature tensor R of (M ; g) is given by

R(X ; Y )Z = k(g(Y ;Z )X � g(X ;Z )Y ):Definition 1.50. Let (M ; g) be a Riemannian manifold, p 2 M andfe1; : : : ; emg be an orthonormal basis of TpM . Then

i) the Ricci tensor at p 2 M Ricp(X ) is defined by

Ricp(X ) = mXi=1

R(X ; ei)ei;ii) the Ricci curvature at p 2 M Ricp(X ; Y ) by

Ricp(X ; Y ) = mXi=1

g(R(X ; ei)ei; Y ); and

iii) the scalar curvature s(p) bys(p) = mXj=1

Ricp(ej; ej) = mXj=1

mXi=1

g(R(ei; ej)ej; ei):Corollary 1.51. Let (M ; g) be a Riemannian manifold of constant sectional

curvature k. Then the following holdss(p) = m � (m� 1) � k:16

PROOF. Let fe1; : : : ; emg be an orthonormal basis, then Lemma 1.49 im-plies that

Ricp(ej; ej) = mXi=1

g(R(ej; ei)ei; ej)= mXi=1

g(k(g(ei; ei)ej � g(ej; ei)ei); ej)= k(

mXi=1

g(ei; ei)g(ej; ej)� mXi=1

g(ei; ej)g(ei; ej))= k(

mXi=1

1� mXi=1

dij) = (m� 1) � k:To obtain the formula for the scalar curvature s we only have to multiply theconstant Ricci curvature Ricp(ej; ej) by m. �

The following result is called the 2nd Bianchi identity.

Lemma 1.52. Let (M ; g) be a Riemannian manifold and R(X ; Y ) its Rie-mann curvature tensor, then

(rXR)(Y ;Z ) + (rYR)(Z ;X ) + (rZR)(X ; Y ) = 0;for all X ; Y ;Z 2 C1(TM ).

17

CHAPTER 2

Structures on Tangent Bundles

In this chapter we introduce necessary structures on the tangent bundleTM , some of which were introduced in Chapter 1 for a differentiable manifoldM . We start with the vertical and the horizontal lifts of vector fields. Theseare then used to calculate the Lie bracket on TM . We then introduce theimportant class of natural metrics on the tangent bundle and calculate theirLevi-Civita connections. For further interest we recommend [21].

1. The Vertical and Horizontal Lifts

Definition 2.1. Let (M ; g) be a Riemannian manifold with Levi-Civitaconnection r, let TM denote the tangent bundle of M , and p be the naturalprojection of TM onto M . Then the differential dp of p is a C1-map fromTTM onto TM . If (p; u) 2 TM , then we denote the kernel of dp at (p; u) byV(p;u) = Ker(dp j(p;u))

and call it the vertical subspace of T(p;u)TM at the point (p; u).

Remark 2.2. If p 2 M and i : TpM ! TM is the inclusion map forthe submanifold TpM = p�1(fpg) in TM with i(u) = (p; u), then for everyu 2 TpM we have V(p;u) = di((TpM )u):

In order to define the very important horizontal subspace we first need tointroduce the connection map K(p;u) from T(p;u)TM to TpM induced by theLevi-Civita connection on (M ; g).

Definition 2.3. Let V be a neighborhood of p in M such that the ex-ponential map expp : TpM ! M maps a neighborhood V 0 of 0 in TpMdiffeomorphically onto V . Furthermore let t : p�1(V ) ! TpM be the C1-map into TpM , which translates every Y 2 p�1(V ) in a parallel manner fromq = p(Y ) to p along the unique geodesic arc in V between q and p. Foru 2 TpM let R�u : TpM ! TpM be the map given by R�u(X ) := X � u forX 2 TpM . Then the connection map

K(p;u) : T(p;u)TM ! TpM

of the Levi-Civita connection r is defined as:

K (A) := d (expp Æ R�u Æ t)(A)

for all A 2 T(p;u)TM .

19

We now establish an important result for the the connection map K .

Lemma 2.4. [6] The connection map K(p;u) : T(p;u)TM ! TpM of r satis-fies

K (dZp(Xp)) = (rXZ )p;where Z 2 C1(TM ) is viewed as a map Z : M ! TM and Xp 2 TpM.

PROOF. Let Z be a vector field which takes the value (p; u) at p. Further-more let g be a geodesic curve such that g(0) = p and _g(0) = Xp. Then thedefinition of the connection map implies

K(p;u)(dZp(Xp)) = d (expp Æ R�u Æ t)(dZp(Xp))= d

dtjt=0expp Æ R�u Æ t(Zg(t))= d

dtjt=0expp(t(Zg(t))� u)= d (expp)0(

d

dtjt=0t(Zg(t)))= d

dtjt=0t(Zg(t)):

In the second last step we use the fact that t((p; u)) = u. If v 2 TpM , then

gp(d

dtjt=0t(Zg(t)); v) = d

dtjt=0gp(t(Zg(t)); v):

Since t is an isometry we have

d

dtjt=0gg(t)(Zg(t); t�1

t v) = d

dtjt=0gg(t)(Zg(t); vt):

The last equation follows from the fact that t is a parallel transport of v alongg. The last term is equal to gp(rXZ ; v), so

d

dtjt=0t(Zg(t)) = rXZ :

This completes the proof. �Let g : I ! M be a curve in M with g(0) = p and g0(0) = u. Let

X : I ! TM be a vector field along g, i.e. a curve in the tangent bundlethrough (p; u) with p Æ X = g. In other words X maps every t to (g(t);U (t))with U (t) 2 Tg(t)M and U (0) = u. Then the connection map K ofr satisfies

K(p;u) : X 0(t) 7! (rg0U )(0):Equipped with the connection map K of r we can now define the horizontalsubspace.

20

Definition 2.5. The horizontal subspace H(p;u) of the tangent spaceT(p;u)TM at (p; u) of the tangent bundle TM is defined byH(p;u) = Ker(K(p;u)):A curve X : I ! TM is said to be horizontal if X 0(t) 2 H(g(t);U (t)) for all t 2 Iand vertical if X 0(t) 2 V(g(t);U (t)) for all t 2 I .

This means that horizontal curves in the tangent bundle TM correspondto parallel vector fields on the manifold (M ; g;r). This is the main motivationfor defining the horizontal space as above.

Proposition 2.6. [16], [6] The tangent space T(p;u)TM of the tangent bundleTM at the point (p; u) is the direct sum of its vertical and horizontal subspaces, i.e.

T(p;u)TM = H(p;u) � V(p;u):PROOF. [6] The map d (expp Æ R�u) an isomorphism of TuTpM , i.e. the

tangent space of TpM at u 2 TpM , and t is defined as the parallel transport ofp. Hence the image of K , which is the horizontal spaceH(p;u) depends only onthe image of p, but the vertical subspace V(p;u) is just the kernel of p. Therefore,by the dimension theorem, the proposition holds. �

With the horizontal and vertical subspaces we can now define the horizon-tal and the vertical lifts of tangent vectors on M .

Definition 2.7. Let X 2 TpM be a tangent vector, then the horizontallift of X at a point (p; u) 2 TM is the unique vector X h 2 H(p;u) such that

dp(X h) = X . The vertical lift of X at (p; u) is the unique vector X v 2 V(p;u)

such that X v(df ) = X (f ) for all functions f on M . Here df is the functiondefined by (df )(p; u) = u(f ).

This can now be extended from tangent vectors to vector fields.

Definition 2.8. The horizontal lift of a vector field X 2 C1(TM ) onTM is the vector field X h 2 C1(TTM ) whose value at a point (p; u) is thehorizontal lift of Xp at (p; u). The vertical lift of a vector field is defined in thesame way. More precisely: If X 2 C1(TM ), then there is exactly one vectorfield X h 2 C1(TTM ) on TM called the horizontal lift of X such that for allZ 2 TM :

dp(X h)Z = Xp(Z ) and KX hZ = 0p(Z ):

The vertical lift X v is the unique vector field satisfying

dp(X v)Z = 0p(Z ) and KX vZ = Xp(Z ):

Note that the maps X 7! X h and X 7! X v are isomorphisms betweenthe vector space TpM and the subspaces H(p;u) and V(p;u), respectively. Each

tangent vector Z 2 T(p;u)TM can then be written as

Z = X h + Y v

21

where X ; Y 2 TpM are uniquely determined by X = dp(Z ) and Y = K (Z ).It follows that if f : M ! R is a smooth real valued function on M , then

X h(f Æ p) = (Xf) Æ p and X v(f Æ p) = 0

for all X 2 C1(TM ).

2. The Lifts in Local Coordinates

In this section we shall express the vertical and the horizontal lifts of vectorfields in terms of local coordinates on M .

Let M be a manifold of dimension m, (x1; : : : ; xm) : U � M ! Rm belocal coordinates on M and define the smooth functions v1; : : : ; v2m : TM !R in the following way:

vi = xi Æ p;vm+i(Y ) = Y (xi) = dxi(Y )

for i = 1; : : : ;m and Y 2 TM . Then (v1; : : : ; v2m) : p�1(U ) � TM ! R2m

are local coordinates on TM . Using Definition 2.8 we now see that

dp((��vi

)Z ) = (��xi

)p(Z ) and dp((��vm+i

)Z ) = 0

for all Z 2 TM and i = 1; : : : ;m, since

dp(��vi

)(f ) = ��vi(f Æ p) = ��xi

(f )

and

dp(��vm+i

)(f ) = ��vm+i

(f Æ p) = ��vm+i

(f (x)) = 0:This leads to the following result for the horizontal and vertical lifts X h;X v ofthe vector field X 2 C1(TM ):

Lemma 2.9. [6] Let (M ; g) be a Riemannian manifold and X 2 C1(TM )be a vector field on M which locally is represented by

X = mXi=1

xi��xi:

Then its vertical and horizontal lifts X v and X h are given by

X v = mXi=1

xi��vm+i

and

X h = mXi=1

xi��vi� (

mXi;j;k=1

xjhkG ijk)

��vm+i;

22

where the coefficients G iab are the Christoffel symbols of the Levi-Civita connectionr on (M ; g).

PROOF. The vertical part follows directly by Definition 2.8.The fact that X h and X are p-related, i.e. dp(X h

Z ) = Xp(Z ), follows directlyfrom

dp((��vm+i

)Z ) = 0 and dp((��vi

)Z ) = (��xi

)p(Z ):Now let the vector field Z 2 C1(TM ) be represented by

Z = mXi=1

hi��xi2 C1(TM ):

Then the map Z : M ! TM is locally given by

Z : M ! TM ; Z : (x1; : : : ; xm) 7! (x1; : : : ; xm; h1; : : : ; hm):Hence

dZ (X ) = dZ (

mXi=1

xi��xi

)= mXi=1

xi��vi

+ mXi;k=1

xi�hk�xi

��vm+k= mXi=1

xi��vi

+ mXk=1

X (hk)��vm+k= mX

i=1

xi��vi

+ mXi=1

X (hi)��vm+i

:Put Xi = ��xi

and let G ijm be the Christoffel symbols of r given by

(rXjXm) = mX

i=1

G ijmXi:

Then we get the following expression for the Levi-Civita connection r.rXZ = mXi=1

rX(hiXi)= mXi=1

(X (hi)Xi + hirXXi)= mXi=1

X (hi)Xi + mXi;j=1

hixjrXjXi

23

= mXi;j=1

X (hi)Xi + mXi;j;k=1

xjhiG kjiXk= mX

i;j=1

X (hi)Xi + mXi;j;k=1

xjhkG ijkXi= mX

i=1

(X (hi) + mXj;k=1

xjhkG ijk)Xi:

It is now a direct consequence of Definition 2.8 and Lemma 2.4 that

K (dZ (X )) = rXZ = mXi=1

(

mXj=1

(xj(�hi�xj

+ mXk=1

G ijkhk)))

��xi:

This implies that K (dZ (X )) = 0, if and only if

X (hi) = � mXj=1

xj

mXk=1

hkG ijk:

This means that rXZ = 0 if and only if dZ (X ) is in the kernel of K . This isequivalent to the fact that dZ (X ) takes of the form

dZ (X ) = mXi=1

xi��vi� mX

i=1

(

mXj;k=1

xjhkG ijk)

��vm+i:

Hence

X h = mXi=1

xi��vi� mX

i=1

(

mXj;k=1

xjhkG ijk)

��vm+i:

This proves the statement for the horizontal lift of X . �Corollary 2.10. [6] For Xi = ��xi

with (x1; : : : ; xm) local coordinates of M

and Ui = ��viwith (v1; : : : ; v2m) local coordinates of TM we obtain

(Xi)v = Um+i and (Xi)

h = Ui � mXj;k=1

(G jik Æ p)vm+kUm+j:

3. The Lie Bracket

In this section we use the vertical and horizontal lifts to calculate the Liebracket on the tangent bundle.

Theorem 2.11. [6] Let (M ; g) be a Riemannian manifold, r be the Levi-Civita connection and R be the Riemann curvature tensor of r. Then the Liebracket on the tangent bundle TM of M satisfies the following:

i) [X v; Y v] = 0,

24

ii) [X h; Y v] = (rXY )v,

iii) [X h; Y h] = [X ; Y ]h � (R(X ; Y )u)v

for all X ; Y 2 C1(TM ) and any point (p; u) in TM.

PROOF. [6] Using the inclusion map in Remark 2.2, we see that thereexist vector fields X 0; Y 0 2 C1(TuTpM ) which are i-related to X v and Y v,respectively i.e.

X v(p;u) = di(X 0

u) and Y v(p;u) = di(Y 0

u)

for all u 2 TpM . Hence we get

[X v; Y v]Z = di([X 0; Y 0]u):By Definition 2.8 we know that K (X v

(p;u)) = Xp for all u 2 TpM . Therefore X 0and Y 0 are right-invariant vector fields on TpM in its capacity as a Lie groupsince X v

(p;u); Y v(p;u) are in V(p;u) and

Ku(di(X 0u+w)) = Ku+w(di(CX 0

u+w))

by the fact that di Æ dt(A) = dt Æ di(A) = A for all A 2 V(p;u+w).

Remark 2.12. That X 0 is a right-invariant vector field means that X 0v+w =

(dRw)v(X0v ) for all v;w 2 TpM with Rw : TpM ! TpM . The right translation

by w defined by Rw : v 7! w + v.

Hence the right-hand side of the formula vanishes, since TpM is an abelianLie group. This proves i).

According to Definition 2.8 we know that dp(X hZ ) = Xp(Z ) and

dp(Y vZ ) = 0. Hence dp([X h; Y v]) = [dp(X h); dp(Y v)] = 0 and therefore

by Definition 2.8, which tells us that dp((rXY )v) = 0, we get

dp([X h; Y v]) = dp((rXY )v);and on the other hand

dp([X h; Y h]) = [X ; Y ]:So we only have to calculate the function K of the right-hand sides in the lasttwo parts of the theorem. To calculate them we will again use our previousabbreviation Xi = ��xi

, where (x1; : : : ; xm) are local coordinates for M . It is

sufficient to calculate both terms just for X ; Y 2 fX1; : : : ;Xmg, because allour functions are linear in every argument. By Corollary 2.10, including theabbreviations in this corollary, and using [Ui;Uj] = 0 and Um+i(vm+j) = dij

for all i; j 2 f1; : : : ; 2mg, with dij the Kronecker symbol, we obtain

[(Xi)h; (Xj)

v] = mXk=1

(G kij Æ p)Um+k:

25

Using the formula for rXY in the proof of Lemma 2.9 and the Definition 2.8we obtain

K ([(Xi)h; (Xj)

v](p;u)) = (rXiXj)p:

This provides us with ii).In the same way as above we will now, observing Ui(vm+j) = 0 for all

i; j 2 f1; : : : ;mg and Ui(G jkl Æ p) = Xi(G j

kl ) Æ p, calculate

[(Xi)h; (Xj)

h] = mXk;l;n=1

fUj(G kil Æ p)� Ui(G k

jl Æ p) + (G nil Æ p)(G k

jn Æ p)�(G njl Æ p)(G k

in Æ p)gvm+l Um+k= � mXk;l=1

(Rklij Æ p)vm+l Um+k:

Therefore, by again using K (Um+i) = Xi we obtain for Z = (p; u) as in Lemma2.9

K ([(Xi)h; (Xj)

h]Z ) = � mXk=1

hkR(Xi;Xj)Xk = �R(Xi;Xj)Z :This proves iii) and completes the proof. �

4. The Natural Almost Complex Structure

We shall now see that the vertical and horizontal lifts of vector fields on themanifold (M ; g) induce a natural almost complex structure J on the tangentbundle TM . It turns out that J is integrable if and only if (M ; g) is flat. Thenatural almost complex structure on the tangent bundle TM was first studiedby Nagano in [14].

Definition 2.13. Let J : TTM ! TTM be the linear endomorphism ofthe tangent bundle of TM characterized by

dp(JA) = �K (A) and K (JA) = dp(A);for all A 2 C1(TTM ). It follows from Definition 2.8 that

J (X h) = X v and J (X v) = �X h;so the endomorphism J : TTM ! TTM satisfies J 2 = �IdTTM , hence itis an almost complex structure on TM . We call it the natural almost complexstructure on TM with respect to g on M .

Definition 2.14. The Nijenhuis tensor N of an almost complex structureJ is defined by

N (A;B) = [A;B] + J [JA;B] + J [A; JB]� [JA; JB]:26

Proposition 2.15. [19], [6] Let (M ; g) be a Riemannian manifold, r bethe Levi-Civitia connection on M and R be the Riemann curvature tensor of r.Furthermore let N be the Nijenhuis tensor of the almost complex strucure J onTM. Then the following hold

K (N (X v; Y v)Z ) = R(X ; Y )Z ; and dp(N (X v; Y v)Z ) = 0

for all X ; Y ;Z 2 C1(TM ).

PROOF. [6] ¿From the Definitions 2.13 and 2.14 we obtain

N (X v; Y v) = [X v; Y v]� J ([X h; Y v] + [X v; Y h])� [X h; Y h]:The first term on the right hand side is zero, hence

K (N (X v; Y v)Z ) = �K (J ([X h; Y v] + [X v; Y h])Z )� K ([X h; Y h]Z )= �dp([X h; Y v]Z + [X v; Y h]Z ) + R(X ; Y )Z= R(X ; Y )Z :Furthermore

dp(N (X v; Y v)Z ) = �dp(J ([X h; Y v] + [X v; Y h])Z )� dp([X h; Y h]Z )= K ([X h; Y v]Z + [X v; Y h]Z )� [X ; Y ]= rXY �rYX � [X ; Y ]= 0: �Corollary 2.16. [19], [6] The almost complex structure J is integrable if and

only if M is flat, i.e. R � 0.

PROOF. [6] If J is integrable, then N � 0 so it is clear that R � 0. If nowR � 0 holds, then it follows that N (X v; Y v) = 0 for all X ; Y 2 C1(TM ).The definition of N gives

N (A;B) = J (N (JA;B)) = J (N (A; JB)) = �N (JA; JB)

for all A;B 2 C1(TTM ). Hence

N (X v; Y v) = N (X v; Y h) = N (X h; Y v) = N (X h; Y h) = 0:This implies N � 0. �

5. Natural Metrics

In this section we introduce the notion of a Riemannian submersion, de-fined by O0Neill in [15]. This leads to the class of natural metrics on thetangent bundle of a given Riemannian manifold (M ; g).

27

Definition 2.17. Let f : (N ; h) ! (M ; g) be a differentiable map betweentwo Riemannian manifolds. Then f is called a submersion if the differentialdfp : TpN ! Tf(p)M of f is surjective at each point p 2 N . Hence for eachq 2 M , f�1(fqg) is a submanifold of N of dimension dim N - dim M . Wedefine Vp as ker(dfp) i.e. the kernel of the differential of f in the tangent space

TpN , and furthermore Hp as Vp? i.e. the orthogonal complement of Vp in

TpN with respect to h.

This leads to the following:

Definition 2.18. A submersion is said to be horizontally conformal if thereexists a positive function l : M ! R+ such that

g(df(X ); df(Y )) = l2h(X ; Y )

for all X ; Y 2 H. If l � 1 then f is said to be a Riemannian submersion.

The well known notions of a Riemannian submersion and vertical andhorizontal spaces make the following definition very natural. It implies thatthe natural projection p : (TM ; g) ! (M ; g) is a Riemannian submersionand that horizontal and vertical vector fields on TM , introduced earlier in thischapter, are orthogonal.

Definition 2.19. Let (M ; g) be a Riemannian manifold. A Riemannianmetric g on the tangent bundle TM of M is said to be natural with respect tog if:

i) g(p;u)

(X h; Y h) = gp(X ; Y ), and

ii) g(p;u)

(X h; Y v) = 0

for all vector fields X ; Y 2 C1(TM ).

We can now use the Koszul formula in Theorem 1.40 to calculate the Levi-Civita connection r for the tangent bundle (TM ; g) equipped with a naturalmetric g with respect to g on M .

Lemma 2.20. Let (M ; g) be a Riemannian manifold and TM be the tangentbundle of M. Then for each (p; u) 2 TM and every natural metric g on TM thecorresponding Levi-Civita connection r satisfies

i) g(rX hY h;Z h) = g(rX Y ;Z );ii) g(rX hY h;Z v) = �1

2g((R(X ; Y )u)v;Z v);

iii) g(rX hY v;Z h) = �1

2g(Y v; (R(Z ;X )u)v);

iv) g(rX hY v;Z v) = 1

2(X h(g(Y v;Z v)) + g(Z v; (rX Y )v)�g(Y v; (rX Z )v));

28

v) g(rX v Y h;Z h) = 1

2g(X v; (R(Y ;Z )u)v);

vi) g(rX v Y h;Z v) = 1

2(Y h(g(Z v;X v))� g(Z v; (rY X )v)�g(X v; (rY Z )v));

vii) g(rX v Y v;Z h) = 1

2(�Z h(g(X v; Y v)) + g(Y v; (rZ X )v)+g(X v; (rZ Y )v));

viii) g(rX vY v;Z v) = 1

2(X v(g(Y v;Z v)) + Y v(g(Z v;X v))�Z v(g(X v; Y v)));

for all X ; Y ;Z 2 C1(TM ).

PROOF. For any vector fields X ; Y ;Z 2 C1(TM ) and i; j; k 2 fh; vgg(rX iY j;Z k) = 1

2(X i(g(Y j;Z k)) + Y j(g(Z k;X i))� Z k(g(X i; Y j)))�g(X i; [Y j;Z k]) + g(Y j; [Z k;X i]) + g(Z k; [X i; Y j])):

i) This is a consequence of Theorem 2.11, Definition 2.19 and the follow-ing calculations

2g(rX hY h;Z h) = (X h(g(Y h;Z h)) + Y h(g(Z h;X h))�Z h(g(X h; Y h)))� g(X h; [Y h;Z h])+g(Y h; [Z h;X h]) + g(Z h; [X h; Y h]))= X (g(Y ;Z )) + Y (g(Z ; Y ))� Z (g(X ; Y ))�g(X h; [Y ;Z ]h) + g(Y h; [Z ;X ]h) + g(Z h; [X ; Y ]h)+g(X h; (R(Y ;Z )u)v)� g(Y h; (R(Z ;X )u)v)�g(Z h; (R(X ; Y )u)v)= 2g(rXY ;Z ):ii) The second part of the lemma is obtained as follows

2g(rX hY h;Z v) = X h(g(Y h;Z v)) + Y h(g(Z v;X h))�Z v(g(X h; Y h))� g(X h; [Y h;Z v])+g(Y h; [Z v;X h]) + g(Z v; [X h; Y h])= �Z vg(X ; Y )� g(X h; [Y h;Z v])+g(Y h; [Z v;X h]) + g(Z v; [X h; Y h]):29

The first term vanishes, because differentiating a horizontal vector field in avertical direction gives zero. The second and third terms also vanish, becausethe Lie bracket of a horizontal vector field and a vertical vector field is vertical,hence

2g(rX hY h;Z h) = g(Z v; [X h; Y h])= g(Z v; [X ; Y ]h � (R(X ; Y )u)v)= �g(Z v; (R(X ; Y )u)v):iii) This is analogous to the proof of part ii).

iv) The Koszul formula gives

2g(rX hY v;Z v) = X h(g(Y v;Z v)) + Y v(g(Z v;X h))�Z v(g(X h; Y v)))� g(X h; [Y v;Z v])+g(Y v; [Z v;X h]) + g(Z v; [X h; Y v])= X h(g(Y v;Z v))� g(X h; [Y v;Z v])+g(Y v; [Z v;X h]) + g(Z v; [X h; Y v]);but the Lie bracket of two vertical vector fields is equal to zero and hence theresult is proven.

v) This is analogous to the proof of part ii)

vi) and vii) are analogous to iv)

viii) This is a direct consequence of the fact that the Lie bracket of twovertical vector fields vanishes. �

Corollary 2.21. Let (M ; g) be a Riemannian manifold and g be a naturalmetric on the tangent bundle TM of M. Then the corresponding Levi-Civitaconnection satisfies

(rX hY h)(p;u) = (rX Y )h(p;u) � 1

2(Rp(X ; Y )u)v

for all X ; Y 2 C1(TM ).

Corollary 2.22. Let K be the sectional curvature of the tangent bundle(TM ; g) equipped with a natural metric g. Then holds

K (X h; Y h) = K (X ; Y )� 3

4k (R(X ; Y )u)v k2;

for any orthonormal vector fields X and Y 2 C1(TM ).

30

PROOF. This follows directly by the formula of O0Neill, provided in [15],page 465 and Corollary 2.21. �

For later purposes we need the following definition and lemma.

Definition 2.23. Let (M ; g) be a Riemannian manifold and let r be theLevi-Civita connection on the tangent bundle (TM ; g), equipped with a natu-ral metric g . Let F : TM ! TM be a differentiable map preserving the fibersand linear on each of them. Then we define the vertical and horizontal lifts F v

and F h by

F (h)v = mXi=1

hiF (�i)v and F (h)h = mX

i=1

hiF (�i)h;

where h =Pmi=1

hi�i 2 p�1(V ) is a local representation of h 2 C1(TM ).

Lemma 2.24. [10] For any vector field X 2 C1(TM ), x = (p; u) 2 TMand h =Pm

i=1hi�i 2 p�1(V ), we have

i) (rX vF v)x = F (Xp)vx +Pm

i=1hirX vF (�i)

v;ii) (rX vF h)x = F (Xp)

hx +Pmi=1

hirX vF (�i)h;

iii) (rX hF v)x = (rX h(F (u)v)x;iv) (rX hF h)x = (rX h(F (u)h)x:

PROOF. Let (x1; : : : ; xm) be a local coordinate of M in a neighborhood Vof p. Then, using the known abbreviation Xi for ��xi

, we have X v �dxi = dxi(X )

for i 2 f1; : : : ;mg. Hence we getrX vF (h)v = mXi=1

rX v(hiF (�i)v)= mX

i=1

X v(hi)F (�i)v + hirX vF (�i)

v= mXi=1

hi(X )F (�i)v + hirX vF (�i)

v= F (Xp)vx + mX

i=1

hirX vF (�i)v:

The second step follows by the product rule. Similarily we calculate:rX vF (h)h = mXi=1

rX v(hiF (�i)h)= mX

i=1

X v(hi)F (�i)h + hirX vF (�i)

h

31

= mXi=1

hi(X )F (�i)h + hirX vF (�i)

h= F (Xp)hx + mX

i=1

hirX vF (�i)h:

For the two last equations of the lemma we use a differentiable curve g :[0; 1] ! M such that g(0) = p and g0(0) = Xp to get a differentiable curve

U Æ g : [0; 1] ! TM such that U Æ g(0) = x and (U Æ g)0(0) = X hx . By

the definition of our functions F v and F h we get F vjUÆg = (F Æ U )vjUÆg andF hjUÆg = (F Æ U )hjUÆg. This proves part iii) and iv) of the lemma. �

32

CHAPTER 3

The Sasaki Metric

This chapter is devoted to the geometry of the tangent bundle (TM ; g) ofa Riemannian manifold (M ; g) equipped with the Sasaki metric g introducedin [16]. This has interested many mathematicans in the last decades, see forexample Dombrowski [6], Kowalski [10], Aso [1] or Musso and Tricerri [13].

1. The Levi-Civita Connection

Definition 3.1. Let (M ; g) be a Riemannian manifold. Then the Sasakimetric g on the tangent bundle TM is the natural metric given by

i) g(x;u)(Xh; Y h) = gp(X ; Y ),

ii) g(x;u)(Xv; Y h) = 0,

iii) g(x;u)(Xv; Y v) = gp(X ; Y ),

for all vector fields X ; Y 2 C1(TM ).

We can now calculate the Levi-Civita connection of the tangent bundlewith respect to g.

Proposition 3.2. [10] Let r be the Levi-Civita connection of (TM ; g)equipped with the Sasaki metric g. Then

i) (rX hY h)(p;u) = (rX Y )h(p;u) � 1

2(Rp(X ; Y )u)v,

ii) (rX hY v)(p;u) = (rX Y )v(p;u) + 1

2(Rp(u; Y )X )h,

iii) (rX vY h)(p;u) = 1

2(Rp(u;X )Y )h,

iv) (rX vY v)(p;u) = 0

for all vector fields X ; Y 2 C1(TM ).

PROOF. i) This is nothing but Corollary 2.21.

ii) The part iii) of Lemma 2.20 and the symmetry rules in Proposition1.46 give us

2g(rX hY v;Z h) = �g((R(Z ;X )u)v; Y v)= �g(R(u; Y )Z ;X )= g(R(u; Y )X ;Z ):33

Part iv) of Lemma 2.20 implies

2g(rX hY v;Z v) = X h(g(Y v;Z v)) + g(Z v; (rXY )v)�g(Y v; (rXZ )v)= X (g(Y ;Z )) + g(Z ;rXY )� g(Y ;rXZ )= g(Z ;rXY ) + g(Y ;rXZ ) + g(Z ;rXY )� g(Y ;rXZ )= 2g(Z ;rXY ):The last important step follows by the definition of a metric connection.

iii) As above we use part v) of Lemma 2.20 and Proposition 1.46 and get

2g(rX vY h;Z h) = g(X v; (R(Y ;Z )u)v)= g(X ;R(Y ;Z )u)= g(R(u;X )Y ;Z ):Part vi) of Lemma 2.20 gives us further

2g(rX vY h;Z v) = (Y h(g(Z v;X v))�g(Z v; (rYX )v)� g(X v; (rYZ )v))= Y (g(Z ;X )� g(Z ;rY X ) � g(X ;rY Z )= g(Z ;rY X ) + g(X ;rY Z )�g(Z ;rY X ) � g(X ;rY Z )= 0:iv) As usual we use Lemma 2.20 to get

2g(rX vY v;Z h) = (�Z h(g(X v; Y v))+g(Y v; (rZX )v) + g(X v; (rZY )v))= �Z (g(X ; Y )) + g(Y ;rZ X ) + g(X ;rZ Y )�g(Y ;rZX ) � g(X ;rZ Y ) + g(Y ;rZ X )+g(X ;rZ Y )= 0

and

2g(rX vY v;Z v) = X v(g(Y v;Z v)) + Y v(g(Z v;X v))�Z v(g(X v; Y v))= X v(g(Y ;Z )) + Y v(g(Z ;X )) � Z v(g(X ; Y ))= 0:34

The last equation we have, because differentiating a horizontal vector field inthe direction of a vertical one gives zero and by the definition of the metricholds g(X ; Y ) = g(X h; Y h). This completes the proof. �

2. The Curvature Tensor

For calculating the Riemann curvature tensor we need the following result,which is a direct consequence of Lemma 2.24.

Corollary 3.3. [10] Let (M ; g) be a Riemannian manifold and let r be theLevi-Civita connection on the tangent bundle (TM ; g), equipped with the Sasakimetric. Let F : TM ! TM be a differentiable map preserving the fibers andlinear on each of them. Then for any x 2 M and h 2 C1(TTM ) we haverX vF (h)v = F (X )vrX vF (h)h = F (X )h + 1

2(R(u;X )F (h))h:

Proposition 3.4. [10] Let (M ; g) be a Riemannian manifold and R be theRiemann curvature tensor of the tangent bundle (TM ; g) equipped with the in-duced Sasaki metric. Then we have the following formulae:

i) R(p;u)(Xv; Y v)Z v = 0;

ii) R(p;u)(Xv; Y v)Z h = (R(X ; Y )Z + 1

4R(u;X )(R(u; Y )Z )�1

4R(u; Y )(R(u;X )Z ))h

p;iii) R(p;u)(X

h; Y v)Z v = �(1

2R(Y ;Z )X + 1

4R(u; Y )(R(u;Z )X ))h

p;iv) R(p;u)(X

h; Y v)Z h = (1

4R(R(u; Y )Z ;X )u + 1

2R(X ;Z )Y )v

p+1

2((rX R)(u; Y )Z )h

p;v) R(p;u)(X

h; Y h)Z v = (R(X ; Y )Z + 1

4R(R(u;Z )Y ;X )u�1

4R(R(u;Z )X ; Y )u)v

p+1

2((rX R)(u;Z )Y � (rY R)(u;Z )X )h

p;vi) R(p;u)(X

h; Y h)Z h = 1

2((rZ R)(X ; Y )u)v

p+(R(X ; Y )Z + 1

4R(u;R(Z ; Y )u)X

35

+1

4R(u;R(X ;Z )u)Y + 1

2R(u;R(X ; Y )u)Z )h

p

for X , Y , Z 2 TpM.

PROOF. [10] i) The first part of the proposition follows directly by the lastpart of Proposition 3.2 and the fact that the Lie bracket of two vertical vectorfields vanishes.

iii) The last part of Proposition 3.2 and the fact that

[X h; Y v] = (rXY )v;by Theorem 2.11, provide us with

R(X h; Y v)Z v = rX hrY vZ v � rY vrX hZ v � r[X h; Y v]Zv= �rY vrX hZ v= �rY v((rXZ )v + F (u)h);

where F : TM ! TM is the linear fiber preserving map

F : u 7! 1

2R(u;Zp)Xp

for any (p; u) 2 TM . By the last part of Proposition 3.2 we know thatrY v(rXZ )v = 0 and according to Corollary 3.3 we haverY vF (u)h = F (Y )h + 1

2(R(u; Y )F (u))h:

Therefore we get

R(X h; Y v)Z v = �rY vrX hZ v= �rY v(rXZ )v + F (u)h= �rY vF (u)h= �F (Y )h � 1

2(R(u; Y )F (u))h= �(

1

2R(Y ;Z )X + 1

4R(u; Y )(R(u;Z )X ))h:

Hence the third part of the proposition is proven.

ii) Using the 1st Bianchi identity in Proposition 1.46

R(X v; Y v)Z h = R(Z h; Y v)X v � R(Z h;X v)Y v;36

we get by using part iii)

R(X v; Y v)Z h = (�1

2R(Y ;X )Z � 1

4R(u; Y )(R(u;X )Z ))h+(

1

2R(X ; Y )Z + 1

4R(u;X )(R(u; Y )Z ))h:

¿From which the statement follows.

iv) As above we now introduce two mappings F1 : TM ! TM andF2 : TM ! TM by

F1(u) 7! 1

2R(u; Yp)Zp

and

F2(u) 7! �1

2R(Xp;Zp)u

and can now write the third part of Proposition 3.2 in the formrY vZ h = F1(u)h:By the definition of the Riemann curvature tensor we obtain

R(X h; Y v)Z h = rX hrY vZ h � rY vrX hZ h � r[X h; Y v]Zh= rX hF1(u)h � rY v((rXZ )h + F2(u)v)� r(rXY )vZ h= (rX(F1(u)))h � 1

2(R(X ; F1(u))u)v�1

2(R(u; Y )rXZ )h � F2(Y )v � 1

2(R(u;rXY )Z )h= (

1

4R(R(u; Y )Z ;X )u + 1

2R(X ;Z )Y )v+1

2((rX R)(u; Y )Z )h:

The last step is only inserting the mappings F1, F2 and the definition of acovariant derivative. The middle step uses Proposition 3.2 and Corollary 3.3.

v) Using part iv) and the 1st Bianchi identity

R(X h; Y h)Z v = R(X h;Z v)Y h � R(Y h;Z v)X h

and we see that

R(X h; Y h)Z v = (1

4R(R(u;Z )Y ;X )u)v + 1

2((rX R)(u;Z )Y )h�(

1

4R(R(u;Z )X ; Y )u)v � 1

2((rY R)(u;Z )X )h

37

+1

2(R(X ; Y )Z � R(Y ;X )Z )v:

Which implies the result.

vi) For the last part we have the following standard calculations

R(X h; Y h)Z h = rX hrY hZ h � rY hrX hZ h � r[X h; Y h]Zh= rX h((rYZ )h � 1

2(R(Y ;Z )u)v)�rY h((rXZ )h � 1

2(R(X ;Z )u)v)�r[X ; Y ]hZ h + r(R(X ; Y )u)vZ h= (rXrYZ )h � 1

2(R(X ;rYZ )u)v�1

2(rXR(Y ;Z )u)v � 1

4(R(u;R(Y ;Z )u)X )h�(rYrXZ )h + 1

2(R(Y ;rXZ )u)v+1

2(rYR(X ;Z )u)v + 1

4(R(u;R(X ;Z )u)Y )h�(r[X ; Y ]Z )h + 1

2(R([X ; Y ];Z )u)v+1

2(R(u;R(X ; Y )u)Z )h= 1

2((rZR)(X ; Y )u)v + (R(X ; Y )Z )h+1

4(R(u;R(Z ; Y )u)X )h + 1

4(R(u;R(X ;Z )u)Y )h+1

2(R(u;R(X ; Y )u)Z )h:

The last step of these calculations follows by the 2nd Bianchi identity, whichtells us that

(rXR)(Y ;Z )u + (rYR)(Z ;X )u + (rZR)(X ; Y )u = 0: �3. Geometric Consequences

Theorem 3.5. [10], [1] (TM ; g) is flat if and only if (M ; g) is flat.

PROOF. [1] It is a direct consequence of Proposition 3.4 that if R � 0

then R � 0. We now assume that R = 0 and calculate the Riemann curvature

38

tensor for three horizontal vector fields at (p; 0):

Rp(X ; Y )Z = R(p;0)(Xh; Y h)Z h = 0:

Therefore (M ; g) is flat. �For the sectional curvatures of (TM ; g) we have

Lemma 3.6. [1] Let (p; u) 2 TM and X , Y 2 C1(TM ) be two orthonor-

mal tangent vectors at p. Let K (X i; Y j) denote the sectional curvature of the planespanned by X i and Y j, i; j 2 fh; vg. Then we have:

i) K (X v; Y v) = 0,ii) K (X h; Y v) = 1

4jR(Y ; u)X j2,

iii) K (X h; Y h) = K (X ; Y )� 3

4jR(X ; Y )uj2,

where jcdotj is the norm induced by g.

PROOF. i) It follows directly from Proposition 3.4 that for two verticalvectors the sectional curvature vanishes.

ii) If one of the vectors is horizontal and the other is vertical, then Propo-sition 3.4 implies

g(R(X h; Y v)Y v;X h) = g(�1

2R(Y ; Y )X ;X )+g(�1

4R(u; Y )(R(u; Y )X );X ):

But the first term vanishes and the result follows by using Proposition 1.46.

iii) If both vectors are horizontal, then Proposition 3.4 and the fact thatthe scalar product of a vertical and a horizontal vector vanishes gives

g(R(X ; Y )Y ;X ) = g(R(X ; Y )Y ;X ) + 3

4g(R(u;R(X ; Y )u)Y ;X )+1

4g(R(u;R(X ; Y )u)X ;X ):

The last term vanishes and hence the lemma is proven. �We can also proof this lemma by the formulae of O0Neill, see page 465 of

[15].

PROOF. We first observe that in our case O0Neill0s tensor T equals zero.Hence the first part is proven, because the sectional curvature on the fibers iszero.

For the second part we observe, that his formula reduces to:

K (X h; Y v) = k AX hY v k2= k (rX hY v)h k2

39

= j12

R(u; Y )X j2:The last part is just Corollary 2.22 and the definition of the Sasaki metric.�Corollary 3.7. [1] If the sectional curvatures of (TM ; g) are bounded, then

(TM ; g) is flat.

PROOF. [1] Let us assume, that TM is not flat. Then it follows by The-orem 3.5 that M is not flat. Hence there exist a point p 2 M and a pairof orthonormal vectors V ;W 2 TpM such that R(V ;W )u 6= 0 for someu 2 TpM . Then we have

K (V h;W h) = K (V ;W )� 3

4jR(X ; Y )uj2:

Since the set of u satisfying this condition is unbounded, K (V h;W h) is un-

bounded from below. Similarily we show that K (V h;W h) is unbounded above,by using the second formula of Lemma 3.6. �

Corollary 3.8. Let (M ; g) be a Riemannian manifold of constant sectionalcurvature k. Then

i) K (X v; Y v) = 0,i) K (X h; Y v) = 1

4k2g(u;X )2;

i) K (X h; Y h) = k� 3

4k2(g(u;X )2 + g(u; Y )2);

for any orthonormal vector fields X ; Y 2 C1(TM ).

Using Lemma 3.6 we obtain the following result for the scalar curvature sof (TM ; g)

Lemma 3.9. [13] Let s be the scalar curvature of g and s be the scalarcurvature of g. Then the following equation holdss = s� 1

4

mXi;j=1

jR(Xi;Xj)uj2;where fX1; : : : ;Xmg are a orthonormal basis for M.

PROOF. Lemma 3.6 provides us with a formula of the sectional curvatures.We only have to add them for an orthonormal basis. This means that forfY1; : : : ; Y2mg, an orthonormal basis on TM with X h

i = Yi and X vi = Ym+i we

get s = 2mXi;j=1

K (Yi; Yj)

40

= mXi;j=1

(K (X hi ;X h

j ) + 2K (X hi ;X v

j ) + K (X vi ;X v

j ))= mXi;j=1

(K (Xi;Xj) � 3

4jR(Xi;Xj)uj2) + 2

mXi;j=1

1

4jR(Xj; u)Xij2:

Therefore we just have to prove that the two sums are equal. But with u =Pmi=1

uiXi we get

mXi;j=1

jR(Xj; u)Xij2= mXi;j;k;l=1

ukul g(R(Xj;Xk)Xi;R(Xj;Xl )Xi)= mXi;j;k;l;s=1

ukul g(R(Xj;Xk)Xi;Xs)g(R(Xj;Xl )Xi;Xs)= mXi;j;k;l;s=1

uk; ul g(R(Xs;Xi)Xk;Xj)g(R(Xs;Xi)Xl ;Xj)= mXi;j;k;l=1

uk; ul g(R(Xj;Xi)Xk;R(Xj;Xi)Xl )= mXi;j=1

jR(Xj;Xi)uj2:In the middle step we use three parts of Proposition 1.46. �

Corollary 3.10. Let (M ; g) be a Riemannian manifold with constant sec-tional curvature k. Thens = (m� 1)k(m� 1

2k k u k2):

PROOF. By Corollary 1.51 we know that s = m(m � 1)k. By Lemma1.49 we therefore obtains = s� 1

4

mXi;j=1

jR(Xi;Xj)uj2= s� 1

4k2

mXi;j=1

jg(Xj; u)Xi � g(Xi; u)Xjj241

= s� 1

4k2

mXi;j=1

(g(Xj; u)2 + g(Xi; u)2 � 2(g(Xi; u)g(Xj; u)g(Xi;Xj))= m(m� 1)k� 1

2k2(m� 1) k u k2 :

In the last step we just split the sum and use the formula for s. �As a direct consequence we get

Theorem 3.11. [13] (TM ; g) has constant scalar curvature if and only if(M ; g) is flat.

Hence by Theorem 3.5 follows

Corollary 3.12. (TM ; g) has constant scalar curvature if and only if thescalar curvature is zero.

By Theorem 3.11 and by section 1.B of [20] we obtain directly

Corollary 3.13. [13] (TM ; g) is locally homogeneous if and only if (TM ; g)is flat.

Where a manifold (M ; g) is said to be locally homogeneous if for each p; q 2M , there exists a neighbourhood V of p, a neighbourhood W of q and a localisometry f : V ! W such that f(p) = q.

In particular, by Theorem 3, page 303 in [9] volume I,

Corollary 3.14. [13] (TM ; g) is locally symmetric if and only if (TM ; g) isflat.

And finally follows by the proof of Lemma 3.9 we have

Corollary 3.15. [13] (TM ; g) is Einstein if and only if (TM ; g) is flat i.e.that (TM ; g) has constant Ricci curvature if and only if (TM ; g) is flat.

For a deeper look on Einstein manifolds we refer to [2].

42

CHAPTER 4

The Cheeger-Gromoll Metric

The Cheeger-Gromoll metric on the tangent bundle TM of a Riemann-ian manifold M has also been of great interest to mathematicans in the lastdecades. This metric was introduced by J. Cheeger and D. Gromoll [3] in1972, but an explicit expression was first given by E. Musso and F. Tricerri[13] in 1988.

1. The Levi-Civita Connection

Definition 4.1. Let (M ; g) be a Riemannian manifold. Then the Cheeger-Gromoll metric ~g is the natural Riemannian metric on the tangent bundle TMsuch that

i) ~g(p;u)(Xh; Y h) = gp(X ; Y ),

ii) ~g(p;u)(Xh; Y v) = 0,

iii) ~g(p;u)(Xv; Y v) = 1

1+r2 (gp(X ; Y ) + gp(X ; u)gp(Y ; u))

for all vector fields X ; Y 2 C1(TM ). Here r denotes juj =pg(u; u).

From now on we set a = 1 + r2 to simplify the notion.

Proposition 4.2. [18] Let ~r be the Levi-Civita connection of TM withCheeger-Gromoll metric ~g. If X ; Y 2 C1(TM ), then for all (p; u) 2 TM

i) ( ~rX hY h) = (rX Y )h � 1

2(R(X ; Y )u)v;

ii) ( ~rX hY v) = 1

2a (R(u; Y )X )h + (rX Y )v;iii) ( ~rX vY h) = 1

2a (R(u;X )Y )h;iv) ( ~rX vY v) = �1a (~g(X v;U )Y v + ~g(Y v;U )X v)+1 + aa ~g(X v; Y v)U � 1a~g(X v;U )~g(Y v;U )U ;

where U 2 C1(TM ) is the canonical vertical vector at (p; u) defined by

U = mXi=1

vm+i(��vm+1

)(p;u);with u = (vm+1; : : : ; v2m).

43

PROOF. [18] i) The first part of this proposition is just the Corollary 2.21.

ii) For the rest we first claim that~g(Y v; (R(Z ;X )u)v) = �1a~g((R(u; Y )X )h;Z h):In fact, by earlier definitions and the skew-symmetry of R, we get~g(Y v; (R(Z ;X )u)v) = 1a (g(Y ;R(Z ;X )u)+g(Y ; u)g(R(Z ;X )u; u))= �1ag(R(u; Y )X ;Z )= �1a~g((R(u; Y )X )h;Z h):Hence by part iii) of Lemma 2.20~g( ~rX hY v;Z h) = �1

2~g(Y v; (R(Z ;X )u)v)= 1

2a~g((R(u; Y )X )h;Z h):By Definition 2.8 and Lemma 2.9

X h(1a ) = 0 and X h(g(Y ; u) Æ p) = g(rX Y ; u) Æ p:

Hence X h(~g(Y v;Z v)) = ~g((rX Y )v;Z v) + ~g(Y v; (rX Z )v) by using the defini-tion of the metric and so we get by part iv) of Lemma 2.20~g( ~rX hY v;Z v) = 1

2(X h(~g(Y v;Z v)) + ~g(Z v; (rXY )v � ~g(Y v; (rXZ )v= ~g((rX Y )v;Z v):

These two equations and Lemma 2.20 complete the proof of the second part.

iii) Similar calculations give us~g(X v; (R(Y ;Z )u)v) = 1a~g((R(u;X )Y )h;Z h)

and so, by part v) of Lemma 2.20, we get~g( ~rX vY h;Z h) = 1

2~g(X v; (R(Y ;Z )u)v)= 1

2a~g((R(u;X )Y )h;Z h):By part vi) of Lemma 2.20

2~g( ~rX vY h;Z v) = Y h(~g(Z v;X v))� ~g(Z v; (rYX )v)� ~g(X v; (rYZ )v)

44

= ~g(Z v; (rYX )v) + ~g(X v; (rYZ )v)�~g(Z v; (rYX )v)� ~g(X v; (rYZ )v)= 0:which proves the third part of the proposition.

iv) With similar calculations as above and part vii) of Lemma 2.20 we getthat~g( ~rX vY v;Z h) = �Z h(~g(X v; Y v)) + ~g(Y v; (rZX )v) + ~g(X v; (rZY )v)= �~g(Y v; (rZX )v)� ~g(X v; (rZY )v)+~g(Y v; (rZX )v) + ~g(X v; (rZY )v)= 0:By using the equation X v(f (r2)) = 2f 0(r2)g(X ; u) we see that

X v(~g(Y v;Z v)) = � 2a2g(X ; u)(g(Y ;Z ) + g(Y ; u)g(Z ; u))+1a (g(X ; Y )g(Z ; u) + g(X ;Z )g(Y ; u));

because a is a function of r2. The equation ~g(X v;U ) = g(X ; u) holds, becausethe canonical vertical vector U is equal to u at the point (p; u) and by thedefinition of the Cheeger-Gromoll metric~g(X v;U ) = 1a (g(X ; u) + g(X ; u)g(u; u)) = aag(X ; u):This fact and part viii) of Lemma 2.20 givea2~g( ~rX vY v;Z v) = a2

2(X v(~g(Y v;Z v)) + Y v(~g(Z v;X v))�Z v(~g(X v; Y v)))= �g(X ; u)(g(Y ;Z ) + g(Y ; u)g(Z ; u))+a2

(g(X ; Y )g(Z ; u) + g(X ;Z )g(Y ; u))�g(Y ; u)(g(Z ;X ) + g(Y ; u)g(X ; u))+a2

(g(Y ;Z )g(X ; u) + g(Y ;X )g(Z ; u))+g(Z ; u)(g(X ; Y )� g(X ; u)g(Y ; u))�a2

(g(Z ;X )g(Y ; u) + g(Z ; Y )g(X ; u))= �(g(X ; u)g(Y ;Z ) + g(Y ; u)g(Z ;X ))+(1 + a)g(X ; Y )g(Z ; u)�g(X ; u)g(Y ; u)g(Z ; u):45

By using the definition of the metric we see that this is equal to what is statedin the proposition. �

2. The Curvature Tensor

Equipped with the Levi-Civita connection we can now relatively easy cal-culate the Riemann curvature tensor of TM . But first we state the followingdirect consequence of Lemma 2.24.

Corollary 4.3. Let (M ; g) be a Riemannian manifold and let ~r be the Levi-Civita connection on the tangent bundle (TM ; ~g), equipped with the Cheeger-Gromoll metric. Let F : TM ! TM be a differentiable map preserving the fibersand linear on each of them, then for any x 2 M and any h 2 C1(TTM ) wehave~rX vF (h)v = F (X )v�1a (~g(X v;U )F (h)v + ~g(F (h)v;U )X v�(1 + a)~g(F (h)v;X v)U + ~g(X v;U )~g(F (h)v;U )U )

and ~rX vF (h)h = F (X )h + 1

2a (R(u;X )F (h))h:Proposition 4.4. [18] Let ~R be the Riemann curvature tensor of TM equipped

with the Cheeger-Gromoll metric ~g. If X ; Y ;Z 2 TpM, then

i) ~R(X h; Y h)Z h = (R(X ; Y )Z )h� 1

4a (R(u;R(Y ;Z )u)X � R(u;R(X ;Z )u)Y�2R(u;R(X ; Y )u)Z )h+1

2((rZR)(X ; Y )u)v;

ii) ~R(X h; Y h)Z v = (R(X ; Y )Z )v+ 1

2a ((rX R)(u;Z )Y � (rY R)(u;Z )X )h� 1

4a (R(X ;R(u;Z )Y )u� R(Y ;R(u;Z )X )u)v�4~g(Z v;U )(R(X ; Y )u)v)+1 + aa ~g((R(X ; Y )u)v;Z v)U ;iii) ~R(X h; Y v)Z h = 1

2a ((rX R)(u; Y )Z )h + 1

2(R(X ;Z )Y )v� 1

4a ((R(X ;R(u; Y )Z )u)v

46

�2g(Y ; u)(R(X ;Z )u)v)+1 + a2a ~g((R(X ;Z )u)v; Y v)U ;

iv)~R(X h; Y v)Z v = � 1

2a (R(Y ;Z )X )h+ 1

2a2(g(Y ; u)(R(u;Z )X )h�g(Z ; u)(R(u; Y )X )h)� 1

4a2(R(u; Y )R(u;Z )X )h;

v)~R(X v; Y v)Z h = 1a (R(X ; Y )Z )h+ 1

4a2(R(u;X )R(u; Y )Z � R(u; Y )R(u;X )Z )h+ 1a2

(g(Y ; u)(R(u;X )Z )h � g(X ; u)(R(u; Y )Z )h;vi)~R(X v; Y v)Z v = a+ 2a2

(~g(X v;Z v)g(Y ; u)U � ~g(Y v;Z v)g(X ; u)U )+1 + a+ a2a2(~g(Y v;Z v)X v � ~g(X v;Z v)Y v)+a+ 2a2

(g(X ; u)g(Z ; u)Y v � g(Y ; u)g(Z ; u)X v):PROOF. i) With standard calculations we get:~R(X h; Y h)Z h = ~rX h~rY hZ h � ~rY h~rX hZ h � ~r[X h; Y h]Z

h= ~rX h((rY Z )h � 1

2(R(Y ;Z )u)v)� ~rY h((rX Z )h � 1

2(R(X ;Z )u)v)� ~r[X ; Y ]h � (R(X ; Y )u)vZ h= (rXrYZ )h � 1

2(R(X ;rYZ )u)v� 1

4a (R(u;R(Y ;Z )u)X )h � 1

2(rXR(Y ;Z )u)v�(rYrXZ )h + 1

2(R(Y ;rXZ )u)v+ 1

4a (R(u;R(X ;Z )u)Y )h + 1

2(rYR(X ;Z )u)v�(r[X ; Y ]Z )h + 1

2(R([X ; Y ];Z )u)v

47

+ 1

2a (R(u;R(X ; Y )u)Z )h= (R(X ; Y )Z )h + 1

2((rZ R)(X ; Y )u)v� 1

4a (R(u;R(Y ;Z )u)X � R(u;R(X ;Z )u)Y�2R(u;R(X ; Y )u)Z )h:The last step follows by the 2nd Bianchy identity.

ii) Note for the second formula that, since ~g(p;u)(Xv;U ) = gp(X ; u), we get~g(p;u)((R(X ; Y )u)v;U ) = gp(R(X ; Y )u; u) = 0. Hencea~R(X h; Y h)Z v = a ~rX h~rY hZ v � a ~rY h~rX hZ v � a ~r[X h; Y h]Z

v= ~rX h( 1

2(R(u;Z )Y )h + a(rYZ )v)� ~rY h( 1

2(R(u;Z )X )h + a(rXZ )v)�a ~r[X ; Y ]h � (R(X ; Y )u)vZ v= 1

2(rXR(u;Z )Y )h � 1

4(R(X ;R(u;Z )Y )u)v+1

2(R(u;rYZ )X )h + a(rXrYZ )v�1

2(rYR(u;Z )X )h + 1

4(R(Y ;R(u;Z )X )u)v�1

2(R(u;rXZ )Y )h � a(rYrXZ )v�1

2(R(u;Z )[X ; Y ])h � a(r[X ; Y ]Z )v�(~g((R(X ; Y )u)v;U )Z v + ~g(Z v;U )(R(X ; Y )u)v)+(1 + a)~g((R(X ; Y )u)v;Z v)U�~g((R(X ; Y )u)v;U )~g(Z v;U )U= a(R(X ; Y )Z )v+1

2((rX R)(u;Z )Y � (rY R)(u;Z )X )h�1

4((R(X ;R(u;Z )Y )u)v � (R(Y ;R(u;Z )X )u)v)+� g(Z ; u)(R(X ; Y )u)v+(1 + a)~g((R(X ; Y )u)v;Z v)U :

48

iii) Calculations similar to those above produce the third formula:~R(X h; Y v)Z h = ~rX h~rY vZ h � ~rY v~rX hZ h � ~r[X h; Y v]Zh= 1

2a ~rX h(R(u; Y )Z )h � ~r(rXY )vZ h� ~rY v((rXZ )h � 1

2(R(X ;Z )u)v)= 1

2a ((rXR(u; Y )Z )h � 1

2(R(X ;R(u; Y )Z )u)v)� 1

2a (R(u;rXY )Z )h � 1

2a (R(u; Y )rXZ )h+1

2(R(X ;Z )Y )v � 1

2a~g(Y v;U )(R(X ;Z )u)v� 1

2a~g((R(X ;Z )u)v;U )Y v+1 + a2a ~g((R(X ;Z )u)v; Y v)U� 1

2a~g(Y v;U )~g((R(X ;Z )u)v;U )U= 1

2a ((rX R)(u; Y )Z )h + 1

2(R(X ;Z )Y )v� 1

4a (R(X ;R(u; Y )Z )u)v� 1

2a~g(Y v;U )(R(X ;Z )u)v+1 + a2a ~g((R(X ;Z )u)v; Y v)U :

Here we used Corollary 4.3 to calculate ~rY v(R(X ;Z )u)v.

iv) For the fourth formula note that ( ~rX hU )(p;u) = 0 and X v(p;u)(f (r2)) =

2f 0(r2)gp(X ; u).

2a~R(X h; Y v)Z v = 2a( ~rX h~rY vZ v � ~rY v~rX hZ v� ~r[X h; Y v]Zv)= �2 ~rX h(~g(Y v;U )Z v � (1 + a)~g(Y v;Z v)U+~g(Z v;U )Y v + ~g(Y v;U )~g(Z v;U )U )�a ~rY v

1a (R(u;Z )X )h�2a( ~rY v(rXZ )v � ~r(rXY )vZ v)

49

= �g(Y ; u)(1a (R(u;Z )X )h + 2(rXZ )v)�g(Z ; u)(1a (R(u; Y )X )h + 2(rXY )v)+2ag(Y ; u)(R(u;Z )X )h� 1

2a (R(u; Y )R(u;Z )X )h � (R(Y ;Z )X )h+2(g(Y ; u)(rXZ )v + g(rXZ ; u)Y v�(1 + a)~g(Y v; (rXZ )v)U+g(Y ; u)g(rXZ ; u)U+g(rXY ; u)Z v + g(Z ; u)(rXY )v�(1 + a)~g((rXY )v;Z v)U + g(rXY ; u)g(Z ; u)U= �(R(Y ;Z )X )h � 1

2a (R(u; Y )R(u;Z )X )h+1ag(Y ; u)(R(u;Z )X )h � 1ag(Z ; u)(R(u; Y )X )h:To calculate ~rY v(R(X ;Z )u)h we used Corollary 4.3. For the last equation wehave to show that all the terms not containing the Riemannian tensor R vanish.But ~g(Y v; (rXZ )v)U = 1a (g(Y ;rXZ ) + g(Y ; u)g(rXZ ))U

and therefore the rest becomes�2a (g(Y ;rXZ ) + g(Y ; u)g(rXZ ; u) + g(Z ;rXY ) + g(Z ; u)g(rXY ; u))U :This is now equal to�2aX h(~g(Y v;Z v) + ~g(Y v;U )~g(Z v;U ))U

because the Levi-Civita connection is a metric one. The last terms are equalto zero, because we are differentiating vertical vector fields in the direction ofa horizontal vector fields.

v) Since [X v; Y v] = 0, we only have to calculate ~rX v~rY vZ h.~rX v~rY vZ h = ~rX v1

2a (R(u; Y )Z )h= � 1a2g(X ; u)(R(u; Y )Z )h+ 1

4a2(R(u;X )R(u; Y )Z )h + 1

2a (R(X ; Y )Z )h:50

If we now subtract the analog equation for ~rY v~rX vZ h, we get~R(X v; Y v)Z h = � 1a2g(X ; u)(R(u; Y )Z )h + 1

4a2(R(u;X )R(u; Y )Z )h+ 1

2a (R(X ; Y )Z )h + 1a2g(Y ; u)(R(u;X )Z )h� 1

4a2(R(u; Y )R(u;X )Z )h � 1

2a (R(Y ;X )Z )h:vi) Note that ~g(U ;U ) = r2 and ~rX vU = 1a (X v + ~g(X v;U )U ). The last

statement is true, because~rX vU = ~rX vPm

i=1vm+i(

��vm+i)= mX

i=1

X v(vm+i)��vm+i+ mX

i=1

vm+i~rX v( ��vm+i

):By Lemma 2.9 the first sum is equal to X v and the last sum is, by part iv) ofProposition 4.2, equal to 1a ((1� a)X v + ~g(X v;U )U ). Thena2 ~rX v~rY vZ v = a2 ~rX v

1a (�g(Y ; u)Z v � g(Z ; u)Y v�g(Y ; u)g(Z ; u)U + (1 + a)~g(Y v;Z v)U )= �a2X v(1a )(g(Y ; u)Z v + g(Z ; u)Y v+g(Y ; u)g(Z ; u)U � (1 + a)~g(Y v;Z v)U )�a(g(X ; Y )Z v + g(Y ; u) ~rX vZ v + g(X ;Z )Y v+g(Z ; u) ~rX vY v + g(X ; Y )g(Z ; u)U+g(X ;Z )g(Y ; u)U + g(Y ; u)g(Z ; u) ~rX vU�X v(a+ 1)~g(Y v;Z v)U � (1 + a)X v(~g(Y v;Z v))U�(1 + a)~g(Y v;Z v) ~rX vU )= 2g(X ; u)g(Y ; u)Z v + 2g(X ; u)g(Z ; u)Y v+2g(X ; u)g(Y ; u)g(Z ; u)U � 2(1 + a)g(X ; u)~g(Y v;Z v)U�ag(X ; Y )Z v + g(Y ; u)g(X ; u)Z v + g(Y ; u)g(Z ; u)X v+g(Y ; u)g(X ; u)g(Z ; u)U � (1 + a)g(Y ; u)~g(X v;Z v)U�ag(X ;Z )Y v + g(Z ; u)g(X ; u)Y v + g(Z ; u)g(Y ; u)X v+g(Z ; u)g(X ; u)g(Y ; u)U � (1 + a)g(Z ; u)~g(X v; Y v)U�ag(X ; Y )g(Z ; u)U � ag(X ;Z )g(Y ; u)U

51

�g(Y ; u)g(Z ; u)X v � g(Y ; u)g(Z ; u)g(X ; u)U+2ag(X ; u)~g(Y v;Z v)U+(1 + a)(�2a g(X ; u)g(Y ;Z )U�2ag(X ; u)g(Y ; u)g(Z ; u)U+g(X ; Y )g(Z ; u)U + g(X ;Z )g(Y ; u)U+~g(Y v;Z v)X v + ~g(Y v;Z v)g(X ; u)U )= g(Y ; u)g(Z ; u)X v + (1 + a)~g(Y v;Z v)X v+3g(X ; u)g(Z ; u)Y v � ag(X ;Z )Y v+3g(X ; u)g(Y ; u)Z v � ag(X ; Y )Z v+g(X ; u)g(Y ; u)g(Z ; u)U � (a+ 3)g(X ; u)~g(Y v;Z v)U�g(Y ; u)~g(X ;Z )U � g(Z ; u)~g(X ; Y )U :Therefore we get be taking the difference of ~rX v~rY vZ v and ~rY v~rX vZ v:a2~R(X v; Y v)Z v = g(Y ; u)g(Z ; u)X v + (1 + a)~g(Y v;Z v)X v+3g(X ; u)g(Z ; u)Y v � ag(X ;Z )Y v+3g(X ; u)g(Y ; u)Z v � ag(X ; Y )Z v+g(X ; u)g(Y ; u)g(Z ; u)U � (a+ 3)g(X ; u)~g(Y v;Z v)U�g(Y ; u)~g(X ;Z )U � g(Z ; u)~g(X ; Y )U�g(X ; u)g(Z ; u)Y v � (1 + a)~g(X v;Z v)Y v�3g(Y ; u)g(Z ; u)X v + ag(Y ;Z )X v�3g(Y ; u)g(X ; u)Z v + ag(Y ;X )Z v�g(Y ; u)g(X ; u)g(Z ; u)U + (a+ 3)g(Y ; u)~g(X v;Z v)U+g(X ; u)~g(Y ;Z )U + g(Z ; u)~g(Y ;X )U= (a2 + a + 1)(~g(Y v;Z v)X v � ~g(X v;Z v)Y v)�(a + 2)(g(Y ; u)g(Z ; u)X v � g(X ; u)g(Z ; u)Y v)�(a + 2)(~g(Y v;Z v)g(X ; u)U � ~g(X v;Z v)g(Y ; u)U ):This finishes the proof of this proposition. �

3. Geometric Consequences

In the following let k � k denote the norm with respect to ~g and let~Q(V ;W ) =k V k2k W k2 �~g(V ;W )2

be the square of the area of the parallelogramm with sides V and W for V ;W 2C1(TTM ).

52

Lemma 4.5. [18] Let X ; Y 2 C1(TM ) be two orthonormal vector fields,then

i) ~Q(X h; Y h) = 1,ii) ~Q(X h; Y v) = 1a (1 + g(Y ; u)2),

iii) ~Q(X v; Y v) = 1a2 (1 + g(Y ; u)2 + g(X ; u)2).

PROOF. i) The first part follows directly from the first part of the defini-tion of the Cheeger-Gromoll metric.

ii) The second part is proved by calculations similar to those in the firstpart ~Q(X h; Y v) = ~g(X h;X h)~g(Y v; Y v)� ~g(X h; Y v)2= 1a (1 + g(Y ; u)2):

iii) The last part follows from~Q(X v; Y v) = ~g(X v;X v)~g(Y v; Y v)� ~g(X v; Y v)2= 1a (1 + g(X ; u)2)1a (1 + g(Y ; u)2)�(

1a (g(X ; Y ) + g(X ; u)g(Y ; u)))2= 1a2(1 + g(Y ; u)2 + g(X ; u)2): �

Now let ~G be the (2; 0)-tensor on the tangent bundle TM with~G : (V ;W ) 7! ~g(~R(V ;W )W ;V )

for V ;W 2 C1(TTM ).In [18] Sekizawa made a mistake when calculating the term ~G(X v; Y v).

This has been corrected in Lemma 4.6 below. This means that some of theresults which follow are different from those Sekizawa stated in [18]. Thosewhich are identical to his are marked with the symbol [18] and the rest are not.

Lemma 4.6. Let X ; Y be two orthonormal vector fields in C1(TM ), then

i) ~G(X h; Y h) = K (X ; Y )� 3

4a jR(X ; Y )uj2,

ii) ~G(X h; Y v) = 1

4a2 jR(u; Y )X j2,

iii) ~G(X v; Y v) = 1�aa4 (1 + g(X ; u)2 + g(Y ; u)2) + a+2a3 .

PROOF. i) The first part is proved as followsa~G(X h; Y h) = a~g(~R(X h; Y h)Y h;X h)= a~g((R(X ; Y )Y )h;X h)

53

�1

4(~g((R(u;R(Y ; Y )u)X )h;X h)�~g((R(u;R(X ; Y )u)Y )h;X h)�~g((2R(u;R(X ; Y )u)Y )h;X h)+a

2~g((rZ R(X ; Y )u)v;X h)= ag(R(X ; Y )Y ;X )+3

4g(R(u;R(X ; Y )u)Y ;X ):

The last step of this calculation can be done in this way, because a naturalmetric of a vertical and a horizontal vector field vanishes and the fact that theRiemann curvature tensor is skrew-symmetric. Finally we get by the rules forthe Riemann curvature tensor g(R(u;R(X ; Y )u)Y ;X ) = jR(X ; Y )u)j2. Hencewe have proved this part.

ii) The proof of the second part follows along the same line of calculations.a2 ~G(X h; Y v) = a2~g(~R(X h; Y v)Y v;X h)= �a2~g((R(Y ; Y )X )h;X h)+2a� 1

2a (~g(Y v;U )~g(R(u; Y )X )h;X h)� 1

2a~g(Z v;U )~g((R(u; Y )X )h;X h)�1

4~g((R(u; Y )R(u; Y )X )h;X h)= �1

4g(R(u; Y )R(u; Y )X ;X )= 1

4jR(u; Y )X j2:

Here the last two steps follow from the skrew-symmetry of the Riemann cur-vature tensor.

iii) In the last case we calculatea2 ~G(X v; Y v) = a2~g(~R(X v; Y v)Y v;X v)= (1 + a+ a2)(~g(Y v; Y v)~g(X v;X v)� ~g(X v; Y v)2)+(a+ 2)(~g(X v; Y v)g(Y ; u)g(X ; u) � ~g(Y v; Y v)g(X ; u)2)+(a+ 2)(g(X ; u)g(Y ; u)~g(X v; Y v)� g(Y ; u)2~g(X v;X v))= 1 + a+ a2a2((1 + g(X ; u)2)(1 + g(Y ; u)2)

54

�g(X ; u)2g(Y ; u)2)+a+ 2a (g(Y ; u)2g(X ; u)2 � (1 + g(Y ; u)2)g(X ; u)2)+a+ 2a (g(Y ; u)2g(X ; u)2 � g(Y ; u)2(1 + g(X ; u)2))= 1� aa2(1 + g(X ; u)2 + g(Y ; u)2) + a + 2a : �

We are now able to calculate the sectional curvatures of TM .

Proposition 4.7. Let ~K be the sectional curvature function of the tangentbundle TM equipped with the Cheeger-Gromoll metric ~g and X ; Y be two or-thonormal vector fields in C1(TM ). Then we get

i) ~K (X h; Y h) = K (X ; Y )� 3

4a jR(X ; Y )uj2;ii) ~K (X h; Y v) = 1

4a jR(u;Y )X j21+g(Y ;u)2 ;

iii) ~K (X v; Y v) = 1�aa2 + a+2a 1

1+g(X ;u)2+g(Y ;u)2 :PROOF. The statements follow directly by dividing ~G(X i; Y j) by the~Q(X i; Y j) for i; j 2 fh; vg. �As for the Sasaki metric we can prove formula i) and ii) of the above results

for the sectional curvatures with O0Neill0s formulae.

PROOF. i) The first equation follows by Corollary 2.22 and the definitionof the Cheeger-Gromoll metric.

ii) The second equation reduces to:~K (X h; Y h) � ~g(Y v; Y v) = k AX hY v k2= k ( ~rX hY v)h k2= 1

4a2jR(u; Y )X j2 �

Corollary 4.8. Let (M ; g) be a manifold of constant sectional curvature k.Then

i) ~K (X h; Y h) = k� 3k2

4a (g(u;X )2 + g(u; Y )2);ii) ~K (X h; Y v) = k2g(X ;u)2

4a(1+g(Y ;u)2);

iii) ~K (X v; Y v) = 1�aa2 + a+2a 1

1+g(X ;u)2+g(Y ;u)2 ;for any orthonormal vector fields X ; Y 2 C1(TM ).

The next theorem differs from Sekizawa [18] in the parts ii) and iii). Hehas stated, that that these parts are only non-negative, if k is non-negative, but

55

it is obvious that k has nothing to do with part iii) and in part ii) it appearsonly as a square.

Theorem 4.9. Let (M ; g) be a Riemannian manifold with constant sectional

curvature k. Let ~K be the sectional curvature of the tangent bundle TM equippedwith Cheeger-Gromoll metric ~g and X ; Y 2 C1(TM ) be two orthonormal vectorfields on M. Then

i) ~K (X h; Y h) is non-negative if 0 � k � 4

3,

ii) ~K (X h; Y v) is non-negative,

iii) ~K (X v; Y v) is positive.

PROOF. If we use an orthonormal basis fe1; : : : ; emg of the tangent spaceTpM with two vectors of the basis equal to X and Y , then we get

g(Y ; u)2 + g(X ; u)2 � mXi=1

g(ei; u)2 = juj2 = a� 1 � a:Hence the theorem follows directly by Corollary 4.8. �

Corollary 4.10. [18] If the Riemannian manifold (M ; g) is flat, then theCheeger-Gromoll metric ~g of the tangent bundle TM has non-negative sectionalcurvatures, which are nowhere constant.

We now introduce an orthonormal basis for the tangent space T(p;u)TM ofthe tangent bundle TM at the point (p; u). Let fe1; : : : ; emg be an orthonormalbasis for TpM , the tangent space of M at the point p, with e1 = u

r, where r

again is the norm of u with respect to the Cheeger-Gromoll metric. Then weget an orthonormal basis ff1; : : : ; f2mg for T(p;u)TM by setting

fi = ehi ; fm+1 = ev

1 and fm+j = paevj ;

for i 2 f1; : : : ;mg and j 2 f2; : : : ;mg.The following lemma is a direct consequence of Proposition 4.7.

Lemma 4.11. Let ff1; : : : ; f2mg be an orthonormal basis for the tangent space

T(p;u)TM of the tangent bundle TM at the point (p; u) as above and let ~K be thesectional curvatures of TM. Then the following hold~K (fi; fj) = K (ei; ej)� 3

4a jR(ei; ej)uj2;~K (fi; fm+1) = 0;~K (fi; fm+k) = 1

4jR(u; ek)eij2;~K (fm+1; fm+k) = 3a2;~K (fm+k; fm+l ) = a2 + a+ 1a2

;56

for i; j 2 f1; : : : ;mg and k; l 2 f2; : : : ;mg.

Proposition 4.12. Let ff1; : : : ; f2mg be an orthonormal basis for the tangentspace T(p;u)TM of the tangent bundle TM at the point (p; u) as above and let ~s bethe scalar curvature of TM. Then~s = s+ 2a� 3

2a Xi<j

jR(ei; ej)uj2+m� 1a2(6 + (m� 2)(a2 + a + 1)):

PROOF. By the defintion of the scalar curvature we know that~s = 2mXk;l=1

~K (fk; fl )= mXi;j=1

~K (fi; fj) + 2

mXi;j=1

~K (fi; fm+j) + mXi;j=1

~K (fm+i; fm+j)= mXi;j=1

(K (ei; ej)� 3

4a jR(ei; ej)uj2)+1

2

mXi;j=1

jR(u; ej)eij2+2

mXi=2

3a2+ mX

i;j=2

i 6=j

a2 + a+ 1a2= s+ 2a� 3

2a Xi<j

jR(ei; ej)uj2+m� 1a2(6 + (m� 2)(a2 + a + 1)):

For the third step we only use the results of Lemma 4.11. For the fact thatmX

i;j=1

jR(ei; ej)uj2 = mXi;j=1

jR(u; ej)eij2see the proof of Lemma 3.9. �

Corollary 4.13. Let (M ; g) be a Riemann manifold of constant sectionalcurvature k. Let the tangent bundle TM be equipped with the Cheeger-Gromollmetric ~g. Then the scalar curvature ~s of TM satisfies~s = m� 1

2a2(2a2mk+ (2a� 3)(a� 1)k2a

57

+2(6 + (m� 2)(a2 + a + 1))):PROOF. By Lemma 1.49 and for i 6= j we getjR(ei; ej)uj2 = g(k(g(ej; u)ei � g(ei; u)ej); k(g(ej; u)ei � g(ei; u)ej))= k2(a� 1)(di1 + dj1): �Corollary 4.14. [18] If the base manifold M has constant sectional curvaturek, then its tangent bundle TM with Cheeger-Gromoll metric ~g is not (curvature)

homogeneous.

PROOF. [18] Corollary 4.13 implies that ~s is never constant if k is con-stant. �

For a given m � 2, we are now interested in determining the sign of the

scalar curvature ~Sm(a; k) as a function of (a; k) 2 D = [1;1) � R. Thecontour D0 = f(a; k) 2 Dj ~Sm(a; k) = 0g in the (a; k) plane is determined bythe equation

f (a; k) = 0;with

f (a; k) = a(a� 1)(2a� 3)k2 + 2ma2k+ 2(6 + (m� 2)(1 + a+ a2)):We obtain first, that for any (a; k) 2 D the value of f (a; k) is increasing if

m is increasing and that f (a; 0) is positive.If a 6= 1 and a 6= 3=2 and the descriminant of f (a; k) = 0 is non-negative,

we get the solutionsk� = �ma2 �pm2a4 � a(a� 1)(2a� 3)2(6 + (m� 2)(1 + a+ a2))a(a� 1)(2a� 3):

Which can be used to plot the contour D0.When removing D0 from D the rest falls into three connected components.

The scalar curvature is positive in the component D+ containing the point(1; 0) and negative in the other two.

In Figure 1 we have plotted the contour D0 in the (a; k) plane for the casewhen m = 3.

58

–50

0

50

100

1.5 2 2.5 3 3.5 4a

Figure 1.

We are interested in determining those k 2 R such that S3(a; k) is positive(non-negative) for all a 2 [1;1) i.e. which are the horizontal half-lines whichare completely contained in the component D+. The connected component ofD0 which is contained in the upper halfspace (k > 0) is a graph of the solutionk� given above. It has exactly one minimum C3 > 0. The graph of the othersolution k+, where defined, has exatly one maximum c3 < 0. The family ofhorizontal lines that we are looking for are then parametrized by k 2 (c3;C3)

It is easy to see that for m > 3 we get exactly the same qualitative behaviourof the two solutions k� and k+ as for m = 3. This provides us with thefollowing result:

Theorem 4.15. Let (M ; g) be a Riemannian manifold of dimension m � 3and of constant sectional curvature k. Then there exist real numbers cm < � � � <c3 < 0 and 60 < C3 < � � � < Cm such that the tangent bundle (TM ; ~g)

i) has positive scalar curvature if and only if k 2 (cm;Cm),ii) has non-negative scalar curvature if and only if k 2 [cm;Cm],

59

When m = 2 the descriminant of f (a; k) = 0 is positive everywhere, andthe solutions k� are given byk� = �2a2 � 2

pa4 � 3a(a� 1)(2a� 3)a(a� 1)(2a� 3):

–40

–20

0

20

40

60

80

1.5 2 2.5 3 3.5 4a

Figure 2.

In Figure 2 we have plotted the contour D0 in the (a; k) for m = 2. Whenthis is removed from D the rest falls into four connected components. Thescalar curvature is positive in the component D+ containing the point (2;�30)and negative in the other two. When a > 3=2 both the solutions k� arenegative, approaching 0 as a!1.

This leads to the following result

Theorem 4.16. Let (M ; g) be a surface of constant sectional curvature k.Then there exists a real number C2 � 41 such that the tangent bundle (TM ; ~g)

i. has positive scalar curvature if and only if k 2 [0;C2),ii. has non-negative scalar curvature if and only if k 2 [0;C2].

60

Corollary 4.17. Let (M ; g) be a Riemannian manifold of dimension m andof constant sectional curvature k. Then for large m the real number Cm is approx-

imative (10 + 4p

6)m.

PROOF. For large m the term under the root in the formula for k� isapproximative m2a4. Hence we getk� � �2ma

(a� 1)(2a� 3):

The function a(a�1)(2a�3)

has for a � 1 its minimum atp

6

2. If we put a = p

6

2

in the equation above, we get the result of the corollary. �Remark 4.18. If we compare the formula of Proposition 4.12 for the scalar

curvature of the tangent bundle (TM ; ~g) equipped with the Cheeger-Gromollmetric with the formula of Lemma 3.9 for the scalar curvature of (TM ; g)equipped with the Sasaki metric, we can see, that they do not differ a lot. Thequestion in the end of this master thesis is, if in general the scalar curvature of(TM ; g) equipped with any natural metric can be written ass = s+ A

mXi;j=1

jR(ei; ej)uj2 + B;where A and B are functions on the norm of u.

61

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[2] A. L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebi-ete (3) 10, Springer, 1987.

[3] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegativecurvature, Ann. of Math. 96 (1972), 413-443.

[4] M. del Carmen Calvo and G. G. R. Keilhauer, Tensor Fields of Type (0; 2) on theTangent Bundle of a Riemannian Manifold, Geom Dedicata 71 (1998), 209-219.

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Master’s Theses in Mathematical Sciences 2001:E10

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