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Connections Nicolson, Robert Alexander 1972

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CONNECTIONS by ROBERT ALEXANDER NICOLSON B.Sc, University of Br i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of /l7j^L^mo3xA/) The University of British Columbia Vancouver 8, Canada Date ttfal 7 if 72 [ i i ] Supervisor; Dr„ J. Gamst ABSTRACT The main purpose of this exposition i s to explore the relations between the notions of covariant deriva-tive,, connection, and spray. We begin by introducing the basic definitions and then use a method of Gromoll, Klingenberg and Meyer to show that covariant derivatives and connections on vector bundles are i n a natural one-to-one correspondence. We conclude by showing that on the tangent bundle of a manifold, sprays and "symmetric" connections are in a natural one-to-one correspondence. Although we use a different method, this re-establishes a result of Ambrose, Palais, and Singer. TABLE OF CONTENTS Introduction 1 Chapter 1. COVARIANT DERIVATIVES 6 1.1 Definition 6 1.2 Local Coordinates 7 1.03 Parallel Transport 11 1.4 The Difference Between Two Covariant Derivatives 13 1.5 Covariant Derivatives on Manifolds 14 lo6 Geodesies 16 1.7 Difference and Torsion 17 Chapter 2. BUNDLE CONNECTIONS 21 2.1 Tangent Bundles of Smooth Groups 21 2.2 The Double Structure of TE 31 2.3 Definition of TT 34 2.4 The Kernels of Tf' 35 2.5 Connections on Vector Bundles 39 2.6 The Difference Field of Two Bundle Connections 44 Chapter 3* COVARIANT DERIVATIVES VERSUS 4# BUNDLE CONNECTIONS 3.1 From Connection Forms to Covariant Derivatives 4# 3.2 From Covariant Derivatives to Connection Forms 52 3.3 Differences 54 Chapter 4« SPRAYS „ 55 4.1 Preliminaries 55 4.2 Bundle Connections on Manifolds 57 4.3 Second Order Differential Equations and Sprays 60 -4.4 The Connection Map of a Spray 63 Bibliography 65 Appendix 66 1 INTRODUCTION One of the main objects of interest to the differen-t i a l geometer are the geodesies of a Riemannian manifold. Recall that, i n local coordinates, geodesies are solutions of a system of second order d i f f e r e n t i a l equations? oy(t) + r[^(<r(t)) oj'(t) oj'(t) = o where the \^  . are the cl a s s i c a l Christoffel symbols. To simplify the notation, we introduce, for x i n some coordinate domain, the bilinear t lRn X TRn >TRn given by I^( ei» ej) - 2 ^(x)e^, where {e^ \ denotes the standard basis of lRn„ In these terms, the above dif f e r e n t i a l equation reads: <r" ( t ) + Q - ( t ) ( c r ' U ) , cr»(t)) « 0. Of course, the V\ 's depend on the local coordinates one uses. The basic d i f f i c u l t y encountered by the founders of the theory i s that the H i ' 8 n o t trans-form as the components of a tensor". That i s , the Q •s do not define a bilinear map on tangent spaces. It was Levi-Civita who saw that the bilinear maps 2 P' do have an int r i n s i c meanings they allow one to introduce absolute differentiation and parallel trans-port along curves. We r e c a l l the result: for curves CT~ s • I -» M on a Riemannian manifold M one considers vector f i e l d s to along c r, that i s , " l i f t s " of cr to eu'rves CO : I > TM on the tangent bundle TM of M. Absolute differentiation associates with to a further vector f i e l d K^ co along cr. In local coordinates: ; t i >u> »(t) + C - ( t ) ( CT'Ct), to ( t ) ) . co i s called a parallel family along CT" i f ^.ui = 0. So, i n local coordinates, paral l e l families are the solutions of a homogeneous linear differential equation. Therefore, for each curve cr on M, the parallel families along c r form a vector space P,,- , and evaluation at any t« i n the domain of cr gives a linear isomorphism of P<j-with the tangent space of M at c r ( t 0 ) . Put differently, p a r a l l e l families give a specific way of propagating tangent vectors along curves. In terms of parallel families, t r i s a geodesic i f and only i f t r ' i s a parallel family along 0 ~ . Moreover, one can use "parallel transports" along integral curves of a vector f i e l d X to define a general covariant deriv-3 ative of vector f i e l d s Y with respect to X. Considerable effort has been spent i n the last f i f t y years to find i n t r i n s i c formulations of the various aspects of the foregoing theory. The idea of absolute differentiation leads to the notion of a covariant derivative on a vector bundle; that i s , an operator taking a vector f i e l d X on a manifold M and a section S of a vector bundle E over M to a further section V XS of E. In Chapter One we review the basic facts about covariant derivatives and show how one extracts the analogue of the Christoffel symbols from the formal definition. The problem of formulating the idea of parall e l transport along curves turns out to be more subtle. One might be tempted to specify, for each curve (T on the manifold, the vector space of a l l parallel families along o- . However, this i s not how parallel transport arises i n practice. Moreover, i t i s technically d i f f i -cult to formulate smoothness conditions i n such a context. The way out i s to consider the "infinitesimal" aspect of the situation. In other words, one prescribes for each tangent vector ^ at a point x on a manifold M, the vector space o f . a l l " i n i t i a l velocity vectors" of parallel families along curves which pass through x with "velocity" ^ . Note that parallel families are curves on the tangent bundle, hence their "velocity vectors" are i n the tangent bundle of the tangent bundle' So, one i s forced to consider the "double tangent bundle" of a manifold. Actually, one gains i n c l a r i t y by generalizing to the case of an arbitrary vector bundle. Accordingly, i n Chapter Two we start by analyzing the structure of the tangent bundle TE of a vector bundle E over a manifold. The d i f f i c u l t i e s with the failure of the C r i s t o f f e l symbols to "transform like a tensor" show up again: TE i s not a vector bundle over M. TE does, however, carry two distinct vector bundle structures, one over E, and the other over the tangent bundle of M. We then give the formal definition of a connection  on a vector bundle E over a manifold M: i t i s a map which assigns to each tangent vector ^ of M a subspace of the fibre of TE over ^ with respect to the vector bundle structure over TM. Finally, Ambrose, Palais and Singer showed how one can deal dire c t l y with geodesies by introducing the notion of a spray on a manifold Ms i t i s a vector f i e l d on the tangent bundle bundle of M whose integral curves "look like geodesies". In Chapter Four we r e c a l l the basic properties of sprays and show how one obtains the analogue of the Christoffel symbols. The main purpose of this exposition i s to explore the relations between the notions of covariant derivat-5 ive, connection, and spray. In Chapter Three we use a method of Gromoll, Klingenberg, and Meyer to show that covariant deriva-tives and connections on vector bundles are i n a natural one-to-one correspondence. In Chapter Four we show how, on the tangent bundle of a manifold, sprays and "symmetric" connections are i n a natural one-to-one correspondence. Thus, we re-establish the main result of Ambrose, Palais and Singer by a different method. 6 Chapter 1; Covariant Derivatives Section 1.1s Definition Let M be a smooth manifold of dimension n, and let p s E >M denote a smooth vector bundle over M with fibre K k. For an open subset U of M, l e t V(U) denote the vector space of smooth vector f i e l d s defined on U, and ^(E) the vector space of smooth sections of E over U. DEFINITIONS A covariant derivative on E i s an operator V s V(M) X ^(E) > TM(E) ( X , S ) i > Vxs having the following properties: Dl) V X + Y S =V XS +V YS, D2) \7fXs = fVxS, D3) VX(S+T) = VXS + VXT, and D4) V x(fT) = (Xf)T + fVxT, ( where f i s any smooth real-valued function defined on M ). i 7 Section 1.2 s Local Coordinates In order to exhibit local coordinates for covariant derivatives we f i r s t examine their restriction to open subsets of Mo I 1.2.1s Restriction to Open Sets Let M be a smooth manifold of dimension n, and let p % E »M denote a smooth vector bundle on M with k _ fibre B . Let V be a covariant derivative on E, and let U be an open subset of M . LEMMA; (a) I f Y e V(M) vanishes on U, then VyS vanishes .. on U for each S e ^ ( E ) , and (b) i f T € fJj(E) vanishes on U, then \7XT vanishes on U for each X e V ( M ) . PROOFt We w i l l prove only (b), for (a) i s even more straightforward. Fix some point y e U, and l e t f : M * IR be a smooth function, having support i n U, such that f i s identically 1 on some neighbourhood of y. Then by definition fT « 0 € t ( E ) . Thus 0 « V xfT = (Xf)T + f V x T , so, at 7, we have 0 = (Xf)(y)T(y) + f( 7)(V xT)(y) = (V xT)(y). Since y € U was a r b i t r a r i l y chosen, the lemma i s proved, PROPOSITION: If V i s a covariant derivative on E, and U i s an open subset of M, then there exists a covariant derivative V on p~ (U) such that, for any ( X , S ) e V(M) X rjj(E), ( V vS)(y) = V^S(y) for each y « U, where X € V(U) and S e f^CE) are the restrictions of X and S respectively. PROOF: The problem i s to define V on vector f i e l d s and sections which may not be extendable to global vector f i e l d s and sections. So, let X be i n V(U) and S i n H^(E), and f i x some point y € U. Let f be' as i n the proof of the lemma. Then f X e V(M) and fS € [^(E), so we may set ( V j S K v ) * ( V f x f S ) ( y ) . 9 The lemma ensures that V i s well-defined, and by construction i t i s a covariant derivative on p""*( un-satisfying the conclusion of the proposition. § 1.2.2; Explicit Local Coordinates Let M be a smooth manifold of dimension n, and le t p ; E »M denote a smooth vector bundle over M with fibre Et . Let V be a covariant derivative on E, and l e t U be an open subset of M such that TM, the tangent bundle of M, and E are t r i v i a l over U. Note that V(U) may be identified with C^CU^R11), the vector space of smooth ]Rn-valued functions defined on U, and rj(E) with C*(U^!Rk), by considering principal parts. The restricted covariant derivative V thus induces an operator o U : C°°(U^Rn) X C~(U,IRk) > C°°(U,JRk). According to the definition i n [ 1 . 1 ] , this operator SU i s linear over C^ CUjJR) i n the f i r s t argument but not i n the second. I f , however, for F € C^Cu,©11) and G e C^CU^), we define T U(F,G) = SU(F,G) - DG(F) where DG : U > Lin(JRn,]Rk) i s the derivative of G, then one checks easily that T U i s linear over C°°(U,E) i n both variables. 10 LEMMA: If an operator T : C°°(U,IRn) X C°°(U0]Rk) > C~(U,JRk) i s bilinear over C^ CUpIR), then (a) i f F e G^U^IR11) vanishes at x « U, P(F,G) vanishes at x for each G e C°°(U,lRk), and (b) i f G e C°°(U^Rk) vanishes at x € U, P(F,G) vanishes at x for each F e G°°(U,JRn). PROOF: We w i l l prove only the f i r s t assertion. For each i from 1 to n, let E^ € C°° (\J9Bn) denote the constant function onto the i canonical basis vector ofJR n. Then we may write F =s s f.E, i 1 1 where each f^ e C^d^R) vanishes at x. Then for any G e C (U,JRk) we have r(F,G)(x) = rCSfiEiyGXx) = Sf,(x)r(E.,G)(x) i 1 1 = 0. Returning to the earlier discussion, we can now see that for each x e U, induces a bilinear morphism P : JRn X JRk > JRk. 11 That is$ i f , for (dx,u) € IRn X IRk, we choose F 6 C°°(UjIRn) such that F(x) = dx, and G € C°°(U»IRk) such that G(x) = u, and set r^(dx,u) = r U ( F , G ) ( x ) , then, by the lemma, P. i s well-defined; and, by con-struction, i t i s bilinear. Moreover, by definition the map p . TJ > B i i ( m n x m k , m k ) Xl >rx i s smooth. So ? over U, V i s represented by the smooth family of bilinear P s m n X TRk > i n the sense that S U (F,G)(x) = DGl F(x) + T (F(x ) ,G(x) ). |x x Consequently, V-^S(x) i s already determined by the value of X at x and the values of S along any smooth curve f i t t i n g X(x). § 1.3s Paral l e l Transport Let M be a smooth manifold of dimension n, and let p s E >M denote a smooth vector bundle on M with fibre B^. Let V be a covariant derivative on E, and let cr ; I »M be a smooth curve. By a section of E along o~ we mean a smooth 12 such that p « S s c r , The covariant derivative of a  section S of E along c r i s the new section V tS : I — > E of E along c r defined as follows: For fixed t 0 e I, we note that the map c* : t i »-(t6 + t) i s a smooth curve representing c r '(t© ) = t| c ^ c r ( t Moreover, S i s defined along 9 so we may form V^S(t 0 ), which we define to be the value of V^S at t 0 . In local coordinates, i f V i s given by r"1 . T O n v^- 7 D ^ . i r » ^ • kl. UI. M.L , and S by G : I »IRk, one finds that S i s given by t, » G'(t) + r < r ( t ) ( c r , ( t ) , G(t) ). We say that a section of E along c r i s parallel  along c r i f V^S E 0, Locally, finding sections parallel along cr means solving G'(t) + £ ( t ) ( c r ' ( t ) , G ( t ) ) = 0 13 for G. Since t h i s i s a linear homogeneous di f f e r e n t i a l equation, we see that parallel sections along any given curve exist, and are uniquely determined by any one of their values, § 1.4: The Difference Between Two Covariant Derivatives Let M be a smooth manifold of dimension n, and let p : E >M denote a smooth vector bundle over M with fibre E . Let V and V be covariant derivatives on E. Since I^(E) i s a real vector space, the following i s well-definedo Let D(V,V) denote the map D(V,V) : V(M) X ^ ( E ) >^<E) ( x , s ) i > Vxs - Vxs. Clearly D(V,V) satisfies conditions Dl) to D3) of [1.1] and condition D4) yields: D(V,V)(X,fS) = V x(fS)-V x(fS) « (xf)s + fVxs - (xf)s - fVxs • f Vxs - f Vxs Thus the difference D(7,V) i s bilinear over C~(MfR). D(V,V) i s called the difference tensor of V and ^7 . Suppose now that V i s any covariant derivative defined on E, and that D J V(M) X TM(E) -> r M(E) i s bilinear (over C^M,©) ). Setting V « V - D, i t i s t r i v i a l to check that V i s then a covariant 14 derivative of E, and that D(V,V) = D. Locally, over a suitable coordinate domain U i n M9 we may represent V by a bilinear map T u s C~(U,Bn) X C°°(U,Ek) > C°°(U,Rk), and V by a similar map T U. The difference D(V,vT i s then represented by Tu' - which i s again bilinear over C°°(U,IR). Section l'»5i Covariant Derivatives on Manifolds 1 1.5.1s Definition and Local Coordinates Let M be a smooth manifold of dimension n, and let TT s TM >M denote i t s tangent bundle. Recall that this i s a smooth vector bundle over M with fibre IRn. A covariant derivative on M i s a covariant derivative on TM as defined i n [1.1], That i s , an operator V s V(M) X V(M) > V(M) ( X , T ) » > \ 7 X Y satisfying conditions Dl) to D4). Let U be an open subset of M such that rT 1(U) i s t r i v i a l . Then, as i n [1.2.2], we may identify V(U) with C°°(UpIRn). Let X and I be smooth vector f i e l d s on U with corresponding maps F and G € C°°(tT,En) respectively,. Then, i n the notation of [1.2.2]) we have, for x € U, g(F,G)(x) = DG(F)(x) + I^(F(x),G(x) ) where the induced map CI. : 3Rn X En H Rn i s bilinear. Since V i s determined locally by these P *s, we may derive an explicit representation of i n terms of the coordinate system of U. § 1.5.2; Classical Notation We w i l l continue with the notation of the last paragraph. Let E, denote the i canonical basis vector of JRn. Then we may write F = n , E . . i i where each € C^CUjB), and similarly G = 2 g.E.. 3 3 3 Then, since Q i s bilinear, we have I^CFUKGCX)) = Px( 2 f i ( x ) E i , 2 g j(x)E j) i j 2 f±(x)g.U) ^ x(E i,E J.). i t J Moreover, each map rj\ : U > m n ^ ( E ^ E . ) 16 must be smooth, so i t may be written where each € C°°(U,m). Finally we have r(F(x),G( X)) = 2 f t(x)g.(x)[ 2 rt(x)E.], x 1, j 1 J .k 1 J K so r , and hence V, i s given locally by specifying n 3 smooth real-valued functions on U. Section 1.6: Geodesies Let M be a smooth manifold of dimension n, and let V be a covariant derivative on M (that i s , on the tangent bundle of M). Let cr : I ^ M "be a smooth curve. By a vector f i e l d along cr we mean a section to : I > TM of TM along cr. As i n [1.3] we say that oo i s parallel  along cr i f ^U> = 0. Note that the canonical l i f t of CT i s a particular vector f i e l d or ' : I * TM along cr . We say that CT" i s a geodesic with respect to V i f G~ * i s pa r a l l e l along 0~~. So, i n local coordinates, where V i s represented by Tx : E n X l n >Bn, cr i s a geodesic i f and only i f 0 - " ( t ) +IJ ( t )(crHt),cr»(t)) £ 0. In other words, to find geodesies means to solve an 17 expli c i t second order d i f f e r e n t i a l equation which i s quadratic i n 0*'. By the existence theory of ordinary d i f f e r e n t i a l equations, for each x e M, and each <f e T M.,„ there exists a unique geodesic cr • I > M such that cr (0) = x, ando-'(O) = cj . Section 1.7: Difference and Torsion < « Let M be a smooth manifold of dimension n, and l e t V and V be covariant derivatives defined on M. As i n [1.3It their difference tensor D : V(M) X V(M) > V(M) ( X , Y ) i > V XY ~ V XY i s bilinear. In this case, moreover, we may decompose D into symmetric and alternating parts. Thus we write D(X,Y) = S(X,Y) + A(X,Y) where S(X,Y) = (1/2)[D(X,Y) + D(Y,X) ], and A(X,Y) = (1/2)[D(X,Y) - D(Y,X)]. PROPOSITION: The following are equivalent: (1) v7 and \7 have the same geodesies, (2) V XX a V XX for each X € y-(M), and (3) S, the symmetric part of D, vanishes. PROOF; Clearly, (3) implies (2). Conversely, i f 0 = D(X,X) = S(X,X) for a l l X e V(M), we obtain 0 = S(X+Y,X+Y) = 2S(X,Y) for a l l X,Y e V(M), since S i s symmetric. Thus, (2) and (3) are equivalent. To see that (l) and (2) are equivalent, we work i n local coordinates. By definition, a curve cr i s a geo-desic for V i f cr"(t) + r c r ( t ) ( 0-'(t),cr'(t)) = 0, and a geodesic for V i f c r " ( t ) + f j r ( t ) ( c r U t ) , cr'.(t)) = 0 (where T and T represent V and V ) • Since, for each x € M and ^ € TXM, there i s a geodesic cr with tf(0) = y, and cr*(o) = <| we see that (1) means: [^(uju) = r^(u,u) for a l l u €IRn. Consequently, (l) i s equivalent to (2). By the torsion tensor of a covariant derivative V on M, we mean the map T y : V(M) X V(M) > V(M) defined by T 7 (X,Y) = V 2Y - V YX - [X,Y], 19 where [X,Y] i s the usual bracket of smooth vector f i e l d s . It i s easily verified that T v i s bilinear and alternating over C 0 0 (M,JR). A covariant derivative V on M i s said to be torsion-free i f Ty s 0. Locally, i f X and Y i n V(M) are represented by F and G i n C°° (-U,-En) respectively, then [X,Y] i s represented by DG(F) - DF(G). Since V^Y i s represented by DG(F) +rU(F,G), then we know T v (X,Y) w i l l be given by the map X 1 U » JRn x l >rx(G(x),F(x)) - P x(F(x),G(x)). Note that V i s torsion-free i f and only i f each corres-ponding P. i s symmetrico By straightforward computation, one may establish the following? LEMMA; Let V and V be covariant derivatives on M with torsion tensors and Ty respectively. Then i f A denotes the alternating part of the difference tensor of V and V , we have T v - T^ = 2A. PROPOSITION: For any covariant derivative V on M, there exists a unique torsion-free covariant derivative V on M having the same geodesies as V . PROOF; If D = S + A denotes the difference tensor of V and V , then the conditions we want are T£ « 0 and D = A. Therefore, by the lemma, we must set V = V - (1/2)T V . One may show by computation that V i s the desired derivative. 21 Bundle Connections From the geometric viewpoint, i t i s desirable to characterize connections i n terms of morphisms of vector bundles. A detailed digression on the vector bundles involved i s required i n order to accomplish this. We begin by considering smooth groups and their associated tangent bundles. Section 2.1s Tangent Bundles of Smooth Groups A smooth group i s a smooth manifold having a compat-ible group structure. That i s , a structure under which multiplication and the taking of inverses are smooth Operations, we w i l l show that the tangent bundle of a smooth group inherits a compatible group structure. The proof of th i s i s greatly simplified i f we express the definition of a smooth group i n the language of diagrams. § 2.1.1s Definition of a Smooth Group DEFINITIONS A smooth group i s a smooth manifold G together with smooth maps in : G X G * G and i : G * G 22 such that? (l) the following diagram commutes (associativity): G X G X G i d Q X -» G X G m X id G G X G m m -> G (2) there exists a smooth map *• of the one-point manifold into G such that the following diagram commutes (unit element): ( i d Q , * 0 -> G X G (*,IdG) id G G X G -> G , and (3) the following diagram commutes (inverses): ( i d Q , i ) * G X G m G. Note that a l l of the maps appearing i n this d e f i n i -tion are smooth. We may therefor apply the tangent functor T throughout the definition and thus gain infor-23 matIon about the tangent bundle TG. § 2 , 1 , 2 ; The Tangent Group of a Smooth Group Let G be a smooth group and TG the associated tangent bundle. The following theorem shows that TG has a natural group structure compatible with i t s manifold structure, THEOREM: If multiplication and inverses for G are given by maps m and- i respectively, then the tangent maps Tm and Ti induce a compatible group structure on TG, Examining the diagrams of [ 2 , 1 . 1 ] , we see that this i s an easy consequence of the fact that the tangent functor T commutes with products. Thus we need only prove the following lemma. LEMMA: If M and N are smooth manifolds, then T(M X N) and TM X TN are naturally diffeomorphic. PROOF: Let pr^ denote the canonical projection of M X N onto M, and p r 2 the corresponding projection onto N. The definition of M X N ensures that these are smooth maps. Thus they w i l l have smooth tangents: Tpr x s T(M X N) >TM and Tpr~ : T(M X N) »TN. 24 Together, these induce a smooth morphism of T(M X N) onto TM X TN Recall that a tangent vector^at x € M may be represented by a smooth curve cr : I >M where I i s an open interval of IR containing 0, C"(0) = x, and cr-'CO) = f . Let o( represent a tangent vector at x € M, and p a tangent vector at y e N. Then, the definition of M X N ensures that there exists a unique curve f at (x,y) € M X N such that pr^° ^ = and p r 2 ° ^  = P This induces a morphism of TM X TN into T(M X N) which i s easily seen to be smooth. This new morphism i s inverse to the one introduced above, so the manifolds are indeed naturally diffeomorphic. We may now examine more closely the structure of TG. Note that the map Tm ! TG X TG *TG i s given l o c a l l y by: Tm: ( (g,u), (h,v) )i >(m(g,h), T .m(u,v) ). I f ©< represents u, and ^ represents v, then T •j1m(u,v) 25 may be represented by T , where $(t) = m( GL(t), £ (t) ). Thus on the tangent level, the multiplication comes from pointwi.se multiplication of curves. Consequently, the unit element, of TG w i l l be that vector i n the f i b r e of TG over the unit element of G which represents the constant curve at that point. That i s , the zero vector. § 2.1.3s Decomposition of TG Let G be a smooth group, and TG the associated tangent bundle. Let p : TG >G denote the canonical projection. The group structure of TG may be more ex-p l i c i t l y viewed under the decomposition to follow. Let m and i represent multiplication and inver-sion on G respectively, and l e t Tm and Ti be their associated tangent maps. From the definition of Tm as given i n [2.1.2] we derive: p o Tm((g,u),(h,v)) = m(g,h) = m(p X p)((g,u),(h,v)). Thus p i s a smooth group homomorphism of TG onto G. Let OQ denote the canonical "zero-section" of TG. That i s , 0 Q : G >TG gi » (g,0). 26 Since 0 d T G i s represented by the constant curve at S g, and tangent multiplication i s essentially pointwise, 0^ i s also a smooth group homomorphism. Moreover, p ° 0 G i s the identity map on G. This situation may be neatly described algebraically i n terms of a semi-direct product. defined as follows. Let H and G be groups, and l e t p be a group homo-morphism of H onto G. Let K denote the kernel of p, and i the inclusion of K into H. Let s be a group homo-morphism of G into H such that p » s = i d ^ . The follow-ing diagram describes the situation: K- > H ( ^ G. P Note that for any h € plh-s.pdi" 1)) = p( h ) p ( h ) ~ 1 = i so h-s-pCh"1) = i(k) for some k € K. Thus h = i(k)s°p(h), and as sets, H = i(K) X s(G). In terms of thi s decomposi-tion, multiplication i s given by: -i(k)sCg). KkOsCg') - i(k)[s(g)i(k')s(g-1)]s(g)s(g«) K i s normal^ so, s(g)i(k *) s(g"*^ ") e i(K), and so the product i s i n the desired form. I f we abuse notation, and l e t g e G denote the action g s i(k)i > s(g)i(k)s(g~ 1), 27 then our formula becomes [i(k)s(g)]-[i(k')s(g')3 - [i(k)gi(k')]-[s(g)s(g')] Algebraically, then, H i s said to be the semi-direct product of G and K relative to the action of G on i(K) defined above. Returning to the earlier discussion and notation, we have that TG i s the semi-direct product of G with ker(p) relative to the action on ker(p) given by conju-gation with elements of OQ. 1 2 .l.lj.: Reinterpretation of TG Let G be a smooth group with multiplication given by a smooth map m. Let e denote the unit element of G. Let TG be the tangent bundle of G, and l e t p and OQ be as defined i n [2.1.3], We now identify ker(p) and the action of OQ mentioned i n the last section. Clearly, as a manifold, ker(p) i s just T (G). Note that, as a vector space, T G has the structure of an additive group, and that there i s a canonical action of G on T eG given as follows: For g 6 G, let int(g) : G • G be the inner automorphism h\ >ghg~ given by g. Since int(g) leaves e fixed, i t s tangent map at e, 28 ad(g) = T e(int(g)) : T @G >TQG must be linear. The resulting homomorphism ad s G >Lin(T eG,T eG) i s knovm as the ad .joint representation. THEOREM; In the above notation; (a) the group structure on ker(p) induced by Tm i s vector space addition in TgG, and (b) the action of G on ker(p) induced by the semi-direct decomposition of TG i s the adjoint representation. PROOF: To see that (a) holds, we r e c a l l that the unit element of TG i s the zero-vector i n TeG. Therefore, since T m : T G X T G »T G e e e e i s linear, Tem(^ , rr\ ) = TQm( ^  , 0) + Tem( 0, ^  ) = c* +^ . (b) follows from the fact that i f ^  € T QG i s represented by a curve cr f then ad(g)(^j) i s represented by the curve t i >gcr(t)g""1. Thus, i f o<v i s the constant curve with image g, we may rewrite the above map as t\— > ^ ( t ) c r ( t ) [ o A ( t ) r 1 But, by definition, this i s the curve representing -^(g^OoU" 1), so we are done. 29 Note that ker(p), with this structure, i s the additive group of the Lie-Algebra of G, denoted L(G). To sum up: i f G i s a smooth group, then TG i s the  semi-direct product of G with the additive group of L(G)  relative to the ad.joint representation. § 2.1.5 s Actions of Smooth Groups on Smooth Manifolds We define here what i s meant by the action of a smooth group on a smooth manifold. Let G be a smooth group with multiplication given by m. Let M be a smooth manifold. We say that G acts smoothly on M i f there exists a smooth map *t ..: . G X M ->M such that: (l) the following diagram commutes: m X i d G X G X M M i d Q X G X M G X M (2) the following diagram also commutes: (*,id M) » G X M -» M, and Clearly, i f G acts smoothly on M, then TG w i l l act 30 smoothly on TM with the tangent action. 2.1.6: Special Case; GL^ The smooth group that we w i l l be interested i n i s GI»k, the group of a l l linear automorphisms of TR . k 2 As an open subset of IR , GL^ has a smooth structure which clearly i s compatible with the group structure. By definition, GL^ acts lin e a r l y (and hence smoothly) on lRk by: GLk X TRk > lRk (A , u) i *Au. As a manifold, the tangent bundle TGLV simply i s GL^ X Mk, where M^  i s the vector space of a l l ( k x k ) -matrices. To identify the group structure of TGL^, we look at the "tangent action" TGLk X IRk X TRk— >TRk X TRk, where we have identified TE k with 1Rk X E k . Clearly the action i s given by: (A,M,u,v)l >(Au,Mu + Av). Therefore, TGLfc can be identified with the subgroup of ^2k c o n s i s t i n g of a l l matrices of the form fA 0" M A A e GI^jM € Mk 31 Note then that p i s given by A 0* M A -» A, and OQ by A A 0 0 A so the semi-direct decomposition i s K A o" " I o ' "A 0 ' A_ _MA~1 1 • . 0 A. (MA" 1^). Section 2.2: The Double Structure of TE The tangent bundle of a smooth vector bundle inherits two smooth vector bundle structures. 1 2.2 .1 : The Standard Structure Let M be a smooth manifold of dimension n, and let p : E > M denote a smooth vector bundle over M k with fibre TR • Since E i s a smooth manifold, i t has a tangent bundle which we denote n E : TE- »E. We c a l l this the standard vector bundle structure of  TE over E, or simply the E-structure. If the transition function between two intersecting coordinate domains of M i s given by x i >-h(x) where h i s a diffeomorphism between open subsets of 32 TRn, then the corresponding transition for the tangent bundle TM i s given by: (x,dx)i > (h(x) ,h 9(x)dx). Then since local t r i v i a l i z a t i o n s of E have transition functions of the form (x,u)i » (h(x),t(x)u), where t i s a smooth mapping of an open subset of TRn into GL^, the corresponding transitions with respect to the E-structure of TE w i l l be given by (x,u,dx,du) H > (h(x),t(x)u,h'(x)dx,t(x)du+t '(x)(dx)u). Note that "fibrewise", this i s linear in(dx,du). § 2 . 2 . 2 : The Tangent Structure Again let M be a smooth, manifold of dimension n, and let p : E >M denote a smooth vector bundle on M with fibre TR . We w i l l show that the tangent map Tp : TE > TM gives TE a smooth vector bundle structure over TM. It suffices to exhibit a system of local t r i v i a l i z a -tions of TE over TM i n such a way that the transition functions act linearly on the fibres of Tp. To do this, f i x some point i n TE, and suppose that (U, <^  , p ) i s a vector bundle chart at its. image under n-g : TE—>E. 33 Then since p i p°*1(U)^ -» cp(U) X TRk i s a diffeomorphism, Tp s T(p~ 1(U)) >T(c£(U) X IRk) i s also. But we know that the tangent functor T commutes with products, and from the definition of Tp we get that T C p " " 1 ^ ) ) = Tp" 1(TU) ? so (TU,T<j> ,T0) gives a local t r i v i a l i z a t i o n of TE. In [2„2.1] we saw that the transition functions for this system of local coordinates have the form (x,u,dx,du) i > (h(x) ,t(x)u,h '(x)dx,t(x)du+t '(x)(dx)u), where h i s a diffeomorphism of open subsets of TRn, and t i s a smooth map from an open subset of TRn into GL^. If we r e s t r i c t our attention to fibre of Tp over (x,dx), then we induce a morphism of the form (u,du) i * (t(x)u,t(x)du + t'(x)dxu). Note that the matrix representation t(x) 0 t'(x)dx t ( x ) . of this morphism i s identical to the image of the tangent map of t at (x,dx). This l i e s i n TGL^ which was estab-lished i n [2.1.6] to be a subgroup of GL 2 k. So, the transition functions do act linearly on the 34 fibres of Tp, and we hare a vector bundle structure on TE. We c a l l this the tangent structure of TE over TM, or simply the TM-structure. Note that, although the transition functions are linear i n (u,du) and i n (dx,du) separately, they are not linear i n (u,.dx,du), and so we do not get a vector bundle structure for TE over M. Section 2.3? Definition of Tf Let M be a smooth manifold of dimension n, and let p s E •—> M denote a smooth vector bundle over M with fibre B . Let : TM »M denote the tangent bundle of M. We denote by E X TM the submanifold of E X TM con-M sisting of a l l points (^ ,^ ) such that P(?) = i f c ( j ) . Note that i f p*" denotes the projection of E X TM onto E, M then we have a smooth vector bundle over E with fibre JRn, and i f denotes the projection of E X TM onto TM, we have a smooth vector bundle over TM with fibre IR • Local coordinates for E X TM come from the product M structure. In particular, points are denoted l o c a l l y by t r i p l e s (x,u,dx) and the projections by p* : (x,u,dx)i > (x,u) and TTJ^ s (x,u,dx) i > (x,dx). In [ 2 , 2 ] we saw that Tp s TE < »TM and TTe ; TE >E each establish a smooth vector bundle structure on TE. Since the morphisms are smooth, we may combine them to obtain a smooth map (ng,Tp) : TE > E X TM Moreover, by definition, these morphisms agree i n the f i r s t coordinate, so the image of (TT^T ) w i l l be E X TM. E p M Denote thi s new morphism T( : TE >E X TM M As an immediate consequence of the definitions, we see that l o c a l l y T l s (x,u,dx,du) » :—>(x,u,dx). Tf has two interpretations. Considering TE and E X TM as vector bundles over E, TT i s fibre-preserving M and acts linearly fibrewise. Such maps are called E-morphisms. Similarly, considering TE and E X TM M as vector bundles over TM, Tt i s a TM-morphism. Section 2,ki The Kernels of Let M be a smooth manifold of dimension n, and let p s E > M denote a smooth vector bundle on M with 36 fibre IR. Let Tl : TE ~^ E X TM be the "double-M morphism" introduced i n [2.3]. The following commutative diagram summarizes the relationships of the preceding section. E X TM <r M TE Since Tt i s both an E-morphism and a TM-morphism, i t has two distinct kernels. 1 2.4.1: The E-Kernel of TT. Here we consider TT as a vector bundle morphism over E. That i s , we concentrate on the following part of the diagram: TE TT E > E X TM M PROPOSITION: There exists a smooth vector bundle TE i morphism such that i : E X E M E X E M -» TE E X TM M -> 0 i s a short exact sequence of vector bundles over E. Before giving a formal proof of this proposition, 37 we analyze the situation "geometrically". Let V denote the kernel of TT over E, and f i x some e € E. If p(e) = x e M, the fibres of TE and E X TM M over e are T QE and T ^ respectively. So, by definition of Tf , the fibre of ? E over e w i l l be the kernel of the tangent map of p at e. That i s , «= kernel of T Qp : T QE -* T^ M In other words, V". i s the tangent space at e to p™ (x) e —1/ E But p (x) = E i s a vector space, so Y~* may be identi-f i e d with E . We now exhibit the formal proof of the proposition. PROOF: Define i : E X E > TE M loc a l l y by i s (x,u,v)i »(x,u,0,v). Examining the form of the transition maps of the E-structure of TE, we conclude that i i s a vector bundle morphism over E. Moreover, the local description of Tt ensures that i satisfies the conclusion of the proposition. § 2.4.2: The TM-Kernel of Tt . Here we consider Tt as a vector bundle morphism over TM. That i s , we concentrate on the diagram TE » E X TM 3* : TM M TM PROPOSITIONf There exists a vector bundle morphism j i TM X E TE M such that 0 * TM X E — ^ T E — * E X TM — > 0 M M i s an exact sequence of vector bundles over TM. Again, before giving the formal proof, we look at the situation fibrewise. Fix l € TM, and suppose n^C^) = x € M. We f i r s t determine the fibre (TE)^ of TE over <| . By definition (TE)^ = T p " 1 ^ ) . Thinking of TE as the disjoint union of i t s fibres over E we haves (TEL o U (TE)« PIT E = U ' Tp - 1( 6 )'fl T E 5 e«E 5 e eeE 5 e -But Tp"*1^) n T gE i s the inverse image of ^ under T gp. In particular, Tp^Cj) n T eE = p unless e i s over xs that i s , unless e i s i n the fibre E x of E over x. Suppose e i s i n E x. Then Tp*"1(t|) n T gE i s the inverse image of under TQp and hence a translate of 39 the kernel of T o , In [2.4.1] we identified the latter with E . Thus, as a set, (TEL = U (TE) e n T E = U E v = E X E . . J eeE s e e cB x Note that, after this identification, 1f : TE * E X TM M acts on (TE)^ = E x X E x by projection onto the second coordinate. In other words, the fibre of the kernel of Tt over ^ may be identified with E x. Now, the formal proof of the propositions PROOF % Define j s TM X E »TE M loc a l l y by j : (x,dx,u) \ •(x,0,dx,u), and check as before that j has the desired properties. Section 2.5s Connections on Vector Bundles I 2.5.1s Connection Maps Let M be a smooth manifold of dimension n, and let p i E >M denote a smooth vector bundle over M with fibre IRk. Let Tf : TE >E X TM be the "double mor-M phism" of [2.3]. DEFINITIONS A connection map on E i s a smooth section C s E X TM >TE M 40 of TT which i s both an E-morphism and a TM-morphism. In terms of local coordinates, a connection map C must be of the form C : (x,u,dx)I— >(x,u,dx,f(x,u,dx)) where, since C i s an E^raorphism, f acts linearly i n dx, and since C i s a TM-morphism, f acts linearly i n u, when the corresponding fibres are fixed. For each x e M then, C induces a bilinear map which we denote - T • IR nXTR k f c——>TR k by -P s (dx,u) « >f(x,u,dx). The negative sign appearing i n the notation i s intro-duced so as to ensure later agreement with c l a s s i c a l notation. Locally then, a connection map i s given by C : (x,u,dx)i > (x,u,dx, -UJ(u,dx)) where xi > -fZ i s a smooth mapping of an open subset of TRn into the bilinear maps from TR nxm k to TRk. I 2.5.2; Connection Forms There i s an equivalent formulation of [2.5.1] i n terms of connection forms. Let M be a smooth manifold of dimension n, and 41 l e t p s E »M denote a smooth vector bundle on M with fibre IRk. Let C : E X TM , > TE M be a connection map as defined i n [2.5.1], Recall the two short exact sequences of [ 2 . 4 ] , That i s 9 i It E X E » T E — =>E X TM M M over E, and , TM X E — >TE — >E X TM M M over TM, We may associate with C the two retractions? K« : TEi > E X E B M via i„ and Kmf : TE »TM X E M via j . These are given l o c a l l y by Kg s (x,u,dx,du)> >(x,u,du +[^(u,dx) ) and Krpjyj : (XpU,dx ydu)I »(x ?dx ?du +f x(u,dx) ) These agree i n the third coordinate. Thus Kg and K ^ induce the same map K s TE- »E given l o c a l l y by K ;': (x;UpdX;du)i » (x,du + P(u pdx) ). Note that K has the following properties: 42 CF 1 s K i s a vector bundle morphism over p : E — » M whose restriction to V i s projection onto the second coordinate, and CF 2 : K i s a vector bundle morphism over TT^ ? TM—>M TM whose restriction to V i s projection onto the second coordinate, where and V ™ denote the "E-kernel" and "TM-kernel" of TT respectively. DEFINITION; A connection form on E i s a smooth map K s TE >E satisfying conditions CF 1 and CF 2 above. The one-to-one correspondence between connection maps and connection forms i s clear from the construction, so we are done except for some remarks on notation. We denote the connection form corresponding to a connection map C by K C ; and, alternatively, the connec-tion map corresponding to a connection form K by Cg.' Locally then, i f C i s given by C : (x,u,dx)i »(x,u,dx, - r(u,dx) ), then KQ i s given by K C : (x,u,dx,du)» » (x,du +T x(u,dx) ), and i f K i s given by K : (x,u,dx,du)* >(x,du +P(u,dx) ), then C K i s given by C R : (x,u,dx)» »(x,u,dx, ~ x(x,dx) ). 43 § 2„5.3: Alternative Definition of Connection Forms In Chapter Four we w i l l be talking about geodesic sprays, and there a slightly different definition of connection forms w i l l be used. Let M be a smooth manifold of dimension n. Let p : E—>M denote a smooth vector bundle on M with k fibre TR . For any s e TR, let s also denote the smooth map s : E *E given l o c a l l y by: s : (x,u)t »(x,su) Let s^ . denote the corresponding tangent map s . TE • *-TE (x,u,dx,du)i Kx,su,dx,sdu). Then: PROPOSITION: A smooth map K : TE >E i s a connection form on E i f and only i f CF 1 i s satisfied, and: CF 3 : K o s * «s s o K for each s € R. (Refer to [2.5.2] for the definition of connection forms on E, and CF 1.) PROOF: Suppose K : TE s>E i s a smooth map satisfying CF 1 and CF 3. Then, as we have seen, CF 1 implies that l o c a l l y K has the form K : (x,u,dx,du)l »(x,du + [^(ujdx) ) where P ^ U j d x ) i s linear i n dx. By CF 3> P^sujdx) = s l ^ U j d x ) , so, P„(u,dx) i s also linear i n u (cf Appendix). Thus CF 2 of [2o5.2j i s satisfied, so K i s a connection form on E. The converse i s also easily established, so the proposition stands. Section 2 . 6 ; The Difference Field of Two Bundle  Connections The two characterizations of bundle connections i n [ 2 . 5 ] yield two corresponding interpretations for the difference between two connections. § 2 . 6 . 1 ; The Difference Field of Two Connection Maps Let M be a smooth manifold of dimension n, and let p : E »M denote a smooth vector bundle over M with k rv fibre IR . Let C and C be two connection maps on E as defined i n [ 2 . 5 . 1 ] and suppose they are given l o c a l l y by C : (x,u,dx)i >(x,u,dx,-P(u,dx) ) and C % (x,u,dx)i ^(x,u,dx,-P.(u,dx) ). Considering C and C as E-morphisms, their difference ^(CjSf) ; E X TM »TE ^ M - i s given l o c a l l y by Dg(C,C) ; (x,u,dx) r > (x9VLfOt-Vx{\ifdx.) + I"x(u,dx) ). 45 The image of DE(C,8') thus l i e s i n VE, the E-kernel of ff , so under the identification of [2,2.4], we arrive at an induced morphism given locally by (x su,dx)l—^UpUy-P^Uydx) + f^Cujdx) ) € E X E. Similarly, considering C and C as TM-morphisms, - A/. Drj^CCjC) induces a map given loca l l y by (x,u,dx)i >(x,-P(u,dx) + r(u,dx),dx) « E X TM, These maps are both E-morphisms, and agree on the corresponding projections, so they both induce the same map D(C,C) E X TM *E M (x,u,dx) i »(x , -C(u,dx) + n,(u,dx) ). This new morphism we c a l l the difference f i e l d of C and cf. Note that the linear i t y requirements on ' and V ensure D(C,C) acts linearly fibrewise. Suppose now that C i s any connection map on E, and that D : E X TM >E M i s a morphism as above. Then i f C i s given l o c a l l y by and D by C s (x,u,dx) i -> (x,u,dx,-r_(u,dx) ), D : (x,u,dx)i > (x,f x(u,dx) ) , 46 we can define a new connection map C on E by C s (x,u,dx)i —> (x,u,dx,-T (u,dx) + f (u,dx) ). /V It i s easily verified that C i s well-defined, and i s the unique connection map satisfying: D(C,C) = D. I 2.6.2: The Difference Field of Two Connection Forms Let M be a smooth manifold of dimension n, and let p s E—>M denote a smooth vector bundle on M with k . • **» fibre IR . Let K and K be two connection forms on E as defined In [2.5,2], and suppose they are given l o c a l l y by K : (x,u,dx,du) i ^ (x,du +P(u,dx) ) and K : (x,u,dx,du) \ > (x,du * t^Cuydx) ). There are two possible interpretations for the difference between K and K. Considering them as morphisms over p : E >M, their difference DE(K,K) : TE > E (x,u,dx,du)t—>> (x, r.(u,dx)-r.(u,dx) ) X A E i s a vector bundle morphism which annihilates V , since f^(u,0) = u = Tx(u,0) by definition. 47 Passing to the quotient space under the i d e n t i f i c a -tions of [2.5.2] we have an induced morphism given loc-a l l y by (x,u,dx)t ± (x,r v(u,dx)-r.(u,dx)) where (x 0u 9dx) e E X TM. M Similarly, seen as vector bundle morphisms over TT^ t TM *M the difference B m ( K $ ) annihilates TM V . This induces a morphism given locally by (x,u,dx) \ »(x, r c(u,dx )-r ?(u,dx)). That i s , under our identifications, Dg(K,ft;) and Drj^ jCKjK*) induce the same morphism. We c a l l this the difference f i e l d of K and f, and denote i t simply D(K9%). Finally, note that i f C and (if are connection maps, and Kg and K~ are the corresponding connection forms, then D(Kc,K{y) = - 0(0,8:) where the latt e r i s as defined i n [2 e6.1]. The minus sign appearing i n the formula i s the result of the earlier decision to define the P. 's to agree with more clas s i c a l results. 4* Chapter 38 Covariant Derivatives Versus Bundle Connections  Section 3°1* From Connection Forms to Covariant Derivatives § 3»l»ls Tangent Maps of Smooth Sections Let M be a smooth manifold of dimension n, and let k p 8 E >M be a smooth vector bundle on M with fibre H . For a smooth section S 8 M >E of p, we denote by S # : TM ) TE the section TS of Tp. In local coordinates, S i s of the form S : x i >(x,u(x) ) where u i s a smooth mapping of an open subset of B n into E . It follows that S^  i n the corresponding coordinate domains w i l l be given by s (x,dx)\ > (x,u(x) ,dx,u *(x)dx). The follovring property of such morphisms i s t r i v i a l PI s (S + T)^ = S^ + T^ There i s , however, another characteristic property. Let f be any smooth real-valued function defined on M. Note that for a smooth section S of p, fS i s another smooth section. Locally, i f S i s given by S 8 x i > (x,u(x) ) then fS i s given by fS s x i > (x,f(x)u(x) ). 49 The tangent map, (fS)^ , must then be of the form (fS)* : (x,dx)v—Mx,f(x)u(x),dx,f(x)u*(x)+f r(x) (dx)u(x)). This represents a section of TE as a vector bundle over TM, and so the right-hand side may be decomposed to give: (x,f(x)u(x),dx,f(x)u'(x)) + (x,0,dx,f '(x) (dx)u(x)). Clearly, the f i r s t term of this decomposition comes from applying f to S.^  . Examine the second term. Recall from [2.4.2] that the kernel of : TE » E X TM, M considered as a TM-vector bundle morphism i s i : TM X E > TE M . (x,dx,u)» >(x,0,dx,u). The second term of the decomposition l i e s i n the image of this map. This means there exists a global section of pr-^ : TM X E » TM, which we denote dfS, such that locally, for a suitable choice of coordinates, i o dfS : (x,dx) \ > (x,0,dx,f v(x)dx,u(x)). If one abuses notation to identify S with the section (x,dx) i > (x,dx,u(x)) of pr^, (df)S simply i s the product of df : TM > IR with S. 50 This establishes the following property? P2 : (fS)* = fSn -8- i(dfS) i n the foregoing notation. 1 3olo2: The Action of S» on Vector Fields Let M be a smooth manifold of dimension n, and let p s E >M denote a smooth vector bundle on M with fibre TR . Let S and T be smooth sections of p, and denote their tangent maps as i n [3»l.l]. Then i f X and Y are smooth vector f i e l d s on M, and f i s a smooth re a l -valued function defined on M, the following hold: i ) S* -(X + Y) -• S ^ . X + S*.Y, i i ) S* o(fx) = fS*«>X, i i i ) (S + T)* o x = ° X + T^° X. and iv) (fS)^o X «' f S * * X 4- i(XfS) where i i s the injection defined i n [2.4.2]. The proof of the properties i s based on the fact that the following two diagrams commute: TE 51 where TT^ and p E are the vector bundle structural maps of TE over TM and E respectively, and df i s the second component of Tf : TM —TJR ^ f f i X E . 1,3"lo3 ' The Construction Let M be a smooth manifold of dimension n, and let p s E——^M denote a smooth vector bundle over M with k fibre E . Let K : TE >E be a smooth connection form on E as defined i n [2a5«2]„ For a smooth vector f i e l d X on M, and a smooth section S of p, define VXS « K ° S* • X where S# = TS« That i s , the following diagram commutes: M By conditions CF 1 and CF 2 of [2.5v2].f and i ) to i v j of [3»1»2], V i s a well-defined covariant derivative on E, Note that over a fixed coordinate domain i n M, suit-able choices for charts give the following formulas: K s (x,u,dx,du)i »(x,du + HL(u,dx) ), X : x i > (x, | (x)), and S : x i >(x fu(x)): so S^»X : x i > (x,u(x), t| (x) ,u'(x)|(x)), a n d VXS : x i » (x,u •(*) <§ (x) + i^(u(x), <J (x)) 52 S e c t i o n 3»2; From C o v a r i a n t D e r i v a t i v e s t o C o n n e c t i o n  Forms ~ I n [3«1] we e s t a b l i s h e d t h a t c o n n e c t i o n f o r m s i n d u c e c o v a r i a n t d e r i v a t i v e s . We now show t h a t one may go t h e o t h e r way. I 3«-2.1; Uniqueness L e t M be a smooth m a n i f o l d o f d i m e n s i o n n , and l e t p : E »M d e n o t e a smooth v e c t o r b u n d l e o v e r M w i t h f i b r e IR . F i x e e E , and suppose p ( e ) = x e M . L e t S be a smooth s e c t i o n of p s u c h t h a t S(x) = e . T h e n , i n t h e n o t a t i o n o f [3»1]» f o r any smooth v e c t o r f i e l d X on M we have ( S * ° X ) ( x ) € T e E . L e t UV d e n o t e t h e s e t o f e l e m e n t s o f T^E o f t h i s f o r m , e e T h a t i s , ue = ° x ) ( x ) s ( x > = e » x a r b i t r a r y } . R e c a l l , f r o m [2.4»1]» t h a t the d o u b l e morphism Tt : TE v E X TM M E i s a n E - m o r p h i s m h a v i n g a n E - k e r n e l V . We c l a i m t h a t U U vf = T ^ E . e e e I n l o c a l c o o r d i n a t e s , p o i n t s o f TE a r e o f t h e f o r m ( x , u , d x , d u ) | a n d , f p r e = ( x o , u 0 ) , vf i s g i v e n b y : 'vj = { ( x , u ,0,du) du € E k } . S o , we must e s t a b l i s h t h a t U*e c o n t a i n s a l l p o i n t s o f t h e 53 form (x 0,u 0 ,dx,du) with dx ^  Oo Note that, for u 0 and k n du i n IR , and dx / 0 i n K , we can find a smooth map F : TJ > E k , where TJ i s some neighbourhood of x , such that F(x 0) = u, and F'(x 0)dx = du. Now, there exists a vector f i e l d X on M with X(x e) = dx, and a section S of E which agrees with F on a neighbour-hood of x 0, so we are done. APPLICATION; I f K and K are connection forms on E rV Inducing the same covariant derivative, then K = K. PROOF; By definition, K and K agree on V E, and by hypothesis they agree on each U as well. Therefore, by our earlier claim, the result. I 3 . 2 . 2 ; Existence Let M be a smooth manifold of dimension n, and let p : E >M denote a smooth vector bundle on M with fib r e E k . Let V be a covariant derivative on sections of E. We wish to define a connection form on E inducing V as i n [ 3 - 1 ] . By [3 .2 .1] i t suffices to find K locally. Recall that i n local coordinates we may write: VXS s x i *(x,u'(x) c5 (x) + T x(u(x), t| (x))) 54 where -f : IRk X Htn IRk i s bilinear and depends smoothly on x. We define K locally by K s (x,u,dx,du) (x,du + P(u..dx) ). It i s easily verified that the global -morphism K. so. defined satisfied the conditions of a connection form. Section 3.3: Differences Note, i n the foregoing notation, that i f K and K* are connection forms on E inducing covariant derivatives V and V respectively, then the difference f i e l d defined i n [2.6.2], and the difference tensor D(V,V) of [1.3.1] are identical after the usual identi-fication of operators V(M) X r K(E) —> rjj(E) with smooth morphisms TM X E > E M which are linear fibrewise. 55 Chapter 4? Sprays  Section 4ols Preliminaries § 4.1.1? The Double Tangent Bundle Let M be a smooth manifold of dimension n. Denote the tangent bundle of -M" by •tfa s TM : — * M and r e c a l l that i t i s a smooth vector bundle over M with fibre 3Rn. Since TM i s a smooth manifold of dimension n 2, i t has a tangent bundle, called the double tangent bundle of M, which we denote by TTfyj : TTM —-. > TM. Recall that i f a point of M l i e s within the domain of a chart (U, <j?), then we denote i t by i t s image x<= f(U) i n m n . Also, i f (U, <f> ) and ( V , are charts having a non-empty intersection, W, then there i s a coordinate change on W given by a diffeomorphism h : <p(W) ——» *KW)» At the tangent level, points of the corresponding domain are denoted by pairs (x,dx) e <^ (U) X 3Rn, and the co-ordinate change i s given by the diffeomorphism <P(W) x m n — > VL(W) x m n (x,dx) 1 ; > (h(x),h'(x)dx). Since TTM i s the tangent bundle of TM, we w i l l 56 denote points by quadruples (x,dx,x,dx) € (y(U)><dRr^dRn><3En, and the corresponding coordinate change here is given by CD(W) X ffin X l n X f f i n > ^ ( W ) X E N X JR N X E N ( x o d X y X o d x ) t > (h(x),hv(x)dx0hUx)x,h"(x)xdx+hUx)dx) 1 4,1,2: The Double Structure of TTM Let M be a smooth manifold of dimension n. Denote by Tfy * TM >M the tangent bundle of M, and by Uj^ ' TTM » TM the double tangent bundle. Since TM i s a vector bundle over M, TTM w i l l have the double structure discussed i n [2.2], In particular, n m : TTM > TM gives the standard structure of TTM, and we have the following exact sequence of smooth vector bundles over TMs i <ft 0 y TM X TM TTM > TM X TM > 0, M M where, i n local coordinates, ttq^j : (x,dx,x,dx) H > (x,dx), i : (x,dx,x)* »(x,dx,0,x), and Ti J (x,dx,x,dx)i > (x,dx,x). On the other hand, the tangent structure of TTM i s given by: T T ^ : TTM >TM, and another exact sequence over TM i s given by: •4 TM X TM M -> TTM -> TM X TM » 0, M 57 where, again i n local coordinates, Tr^j c (x,dx,x,dx) i > (x,x), and j : (x,dx,x)i >(x,0,x,dx). It i s important to note that these two structures are Isomorphic That i s , If I ^ r ^ denotes the involution of TTM given l o c a l l y by 'TTM s (x,dx,x,dx)i *(x,x,dx,dx), and I T M denotes the canonical involution of TM X TM, TM then the following diagram commutes: i M -* TM X TM M •TM -•TTM I -> TM X TM M •> TTM —> TM X TM M TTM ITM ->0 •^ TM X TM M -»0. Section 4.2: Bundle Connections on Manifolds I 4.2,1: Definition Let M be a smooth manifold of dimension n, with tangent bundle : TM > M. We define a connection map on M to be a connection map on the vector bundle TM as i n [2o5'.l]. That i s , a smooth section C : TM X TM » TTM M of Tf (as defined i n [4<>1»2]) which i s a vector bundle morphism with respect to both the standard and tangent 58 structures of TTM. Similarly, a connection form on M i s defined to be a connection form on TM. That i s , a smooth morphism K s TTM. —-. TM satisfying% CF 1 s K i s a vector bundle morphism whose re s t r i c t i o n to i(TM X TM) i s projection onto the second M coordinate, and CF 2 : K i s a vector bundle morphism over whose res-t r i c t i o n to j ( T M X TM) i s also projection onto M the second coordinate, where i and j are as i n [4e>1.2], In [2,5,3], we saw that condition CF 2 i s equivalent to CF 3 K o s^ = s o K where s e IR denotes fibrewise multiplication by s: s 1 TM y TM (x,dx)i *(x,sdx), and s^ . the tangent map0 § 4.2.2; Connection Maps versus Connection Forms In [3»5] i t was shown that connection maps and connec-tion forms are i n a one-to-one correspondence. Recall that locally, connection maps have the form 59 ! C s (x,dx,x)i > ( x y d X y X p - ^ C d X j x ) ) where Q i s a bilinear mapping of JRn X lR n onto B n depending smoothly on x. Corresponding to each such C we have a connection form K^, given locally by K Q S ( x j d X y X p d x ) t (x,dx + P^C dx,x) ). Since the two bundle structures of TTM are isomorphic, we have a notion of symmetry. That i s , we say that a connection map C on M i s symmetric i f i t commutes with the involutions of TTM and TM X TM introduced i n [4.1-2]. That i s , when c"'° % = I T T M e C o Correspondingly, a connection form K on M i s symmetric when K °  TTM  = K* In local coordinates, this means K and C are symmetric i f and only i f each of the corresponding P *s i s symmet-r i c . Thus C i s symmetric i f and only i f i s symmetric. Finally, r e c a l l that for any two connection maps C and C on M there i s a morphism D(C,C) : TM X TM >TM M called the difference f i e l d of C and C, and that given any 60 connection map C, and a morphism D as stated,, there i s a unique connection map C on M such that D(C,C) = D. No.te that i f C and C are symmetric, then so- i s DtC,C). Section 4.3: Second Order Differential Equations and Sprays § 4.3.1: Second Order Differential Equations Let M be a smooth manifold of dimension n, s TM * M i t s tangent bundle, and T T ^ S TTM >TM i t s double tangent bundle (with standard structure). Recall that Tn^ : TTM TM also gives TTM a smooth vector bundle structure (the tangent structure) and that the canonical involution I : TTM $> TTM i s an isomorphism of these two structures. A second order dif f e r e n t i a l equation on M i s defined to be a smooth vector f i e l d X : TM » TTM which i s a section of both the standard and tangent structures of TTM. That i s , such that n T M o X = i dTM ° T n M ° X ' or equivalently, I 0 X as X . In local coordinates, then, X must have the form X : (x,dx) i > (x,dx,dx, ^  (x,dx)) 61 where ^ i s a smooth IRn~valued map. By definition, a smooth curve p : I > TM i s an integral curve of a vector f i e l d Y on TM i f and only i f p ' ( t ) = Y(^(t)) for each t i n I, where p ' ; I > TTM i s the canonical l i f t of /3 to the tangent bundle of TM. In local coordinates, p may be written t i > (cr ( t ) , t (t)) € U X B ^ Therefore, i f X i s a second order di f f e r e n t i a l equation with local representation (x,dx)i » (x,dx,dx, ^(x,dx)), p i s an integral curve of X i f and only i f s cr'U) = t ( t ) , and t ' ( t ) = <f ( c r ( t ) , t ( t ) ) In other words, the condition i s that X = CT', and cr " « | ( c r , c r ' ) . Therefore, integral curves of second order d i f f e r e n t i a l equations are equal to the canonical l i f t s of their projections onto M. I 4o3.2s Sprays We continue with the notation of the last subsection. Let C : TM >< TM > TTM M 62 be a connection map on M as defined i n [4«2.2], If we r e s t r i c t C to the diagonal of TM X TM, then M the resulting map induces a second order d i f f e r e n t i a l equation . X Q % TM —> TTM called the spray of C. In local coordinates, i f C i s given by C : (x,dx,x)i »(x,dx,x,- (dx,x)) then i s given by XQ : (x,dx)i > (xjdx^dXy-r^CdXydx)). We may characterize those second order d i f f e r e n t i a l equatxons which arise from ccnnectxon maps as fellows! For s € R, l e t s : TM > TM and s s TTM » TTM denote fibrewise multiplication by s, and let s^ . denote the tangent map of s 0 Then i f X^ i s the spray of a connection map C on M, the following condition holds: (Sp) s*<> s * X C = X C ° s, since each Hy. i s quadratic. In general, a smooth morphism X : TM * TTM i s called a spray on M whenever i t i s a second order di f f e r e n t i a l equation on M satisfying condition (Sp). Locally then, i f X i s given by X % (x,dx)l > (x,dx,dx, <S (x,dx)), 63 condition (Sp) implies <^(x,sdx) = s 2^(x,dx), so a second order diffe r e n t i a l equation i s a spray i f and only i f | i s a quadratic form i n dx. Section 4.4 s The Connection Map of a Spray Let M be a smooth manifold of dimension n, and let X s TM — » TTM be a spray on M as defined i n the last section. THEOREM; There i s a unique symmetric connection map C ; TM X TM > TTM M on M generating the spray X. PROOF; Suppose f i r s t that C and C are two symmetric connection maps generating the same spray X. Consider their difference as i n [ 2 . 6 . 1 ] . By hypothesis, i t w i l l vanish on the diagonal of TM X TM, and so must be alter-M nating. On the other hand, i t i s symmetric. Therefore, A/ i t must vanish everywhere. Thus C = C. Since uniqueness i s established, we need only exhibit a suitable map C locally. Suppose X i s given l o c a l l y by X : (x,dx)t > (x,dx,dx, <|x(dx)) where jpx(dx) «= ^(x,dx) i s quadratic i n dx. S i n c e ^ x i s homogeneous of degree two, i t i s the res t r i c t i o n to the diagonal of some bilinear mapping of 3Rn X IRn into lR n (cf Appendix). The family of a l l such maps inducing | x 64 contains a unique symmetric member. Denote this map by -To The following i s then a well-defined connection map: C : TM X TM > TTM M (x,dx,x) i > (x,dx,x,- (dx,x)) and satisfies the hypothesis of the proposition. As a consequence of this theorem, note that given any connection map C on M, there exists a unique symmetric S S connection map C on M such that C and C induce the same spray. Setting, for an arbitrary connection map C on M, T(C) = D(C,CS) : TM X TM * TM, M we obtain an alternating tensor f i e l d T, called the torsion f i e l d of C. Thus, although sprays are not i n one-to-one corres-pondence with bundle connections or covariant derivatives, the explicit formulas i n local coordinates do imply the following relationship. Let V be a covariant derivative on M, let K be the corresponding connection form, and C the corresponding -connection map, and let X be the corresponding spray. Then: (1) V i s torsion-free i f and only i f K and C are symmetric, (2) a smooth curve on M i s a geodesic of V i f and only i f i t i s the projection to M of an integral curve of X, and (3) the torsion tensor of V is equal to the torsion f i e l d of C. 65 BIBLIOGRAPHY 1 . W. AMBROSE, R. S„ PALAIS, I, M. SINGER^ "Sprays", Anais du Academia Brasileira de Ciencias, vol. 32, 163-173 ( I 9 6 0 ) 2 E J . DIEUDONNE: "Elements d'Analyse", tome III, Gauthier-Villars, Paris, ( 1970) 3. Co GODBILLONs "Geometric Differentielle et Mecanique Analytique", Hermann, Paris, ( 1 9 6 9 ) 4. D. GROMOLL, W. KLINGENBERG, W. MEYERs "Riemannsche Geometrie im Grossen", Springer Lecture Notes, Berlin-Heidelberg-New York, 0-968) 5. N. Jo HICKS: "Differential Geometry", Van Nostrand, Toronto-New York-London, ( 1965) 6. S. LANG: "Introduction to Differentiable Manifolds", J* Wiley; New York-London; ( 1 9 6 2 ) 6 6 APPENDIX Homogeneous Smooth Functions Recall that a function f s &n > B*1 i s homogeneous of degree k ^  0, i f and only i f , for each u € B n, (H) f(su) s kf(u) for any s e B. Note that i f f i s homogeneous of degree 0, then (H) implies that f i s a constant function, and i f f i s homogeneous of degree k > 0, then f(0) =0. LEMMA: I f f i s a differentiable homogeneous function of degree k > 0, then Df : B n > Lin(B n, B01) i s homogeneous of degree k-1. PROOF: Differentiating (H) with respect to u gives; sDfl = s kDf| , l(su) lu DfI = s k _ 1 D f I so l(su) l(u) LEMMA: (Euler's Relation) I f f i s a homogeneous differentiable function of degree k > 0, then Df| (u) = kf(u). lu PROOF: Differentiating (H) with respect to s gives l(su) Df I (u) = k s ^ f C u ) . But by the above lemma. Df| (u) o s ^ D f (su) t l ~ (u), u so we are done. As a consequence of these two lemmas, we have the following proposition which may be proved by induction on k, P R O P O S I T I O N : A smooth function F S J R H ^m i s homogeneous of degree k > 0 i f and only i f there exists a k-linear function f : IRn X B n X ..... X ]Rn ?TRm such that for each u € IR11 , £(u,u, .....,u) =s f ( u ) . 

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