CONNECTIONS by ROBERT ALEXANDER NICOLSON B.Sc
CONNECTIONS by ROBERT ALEXANDER NICOLSON
B.Sc, University of British Columbia, 1969
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in 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 it 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 is 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 [ii] Supervisor; Dr„ J. Gamst
ABSTRACT
The main purpose of this exposition is 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 in 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• tial geometer are the geodesies of a Riemannian manifold.
Recall that, in local coordinates, geodesies are solutions of a system of second order differential equations?
oy(t) + r[^( To simplify the notation, we introduce, for x in 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 differential equation reads: 0. not founders of the theory is that the Hi'8 trans- form as the components of a tensor". That is, 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 intrinsic meanings they allow one to introduce absolute differentiation and parallel trans• port along curves. We recall the result: for curves CT~ s • I -» M on a Riemannian manifold M one considers vector fields to along cr, that is, "lifts" of cr to eu'rves CO : I > TM on the tangent bundle TM of M. Absolute differentiation associates with to a further vector field K^co along cr. In local coordinates: ; ti >u> »(t) + C-(t)( CT'Ct), to (t)). co is called a parallel family along CT" if ^.ui = 0. So, in local coordinates, parallel families are the solutions of a homogeneous linear differential equation. Therefore, for each curve cr on M, the parallel families along cr form a vector space P,,- , and evaluation at any t« in the domain of cr gives a linear isomorphism of P In terms of parallel families, tris a geodesic if and only if tr' is a parallel family along 0~. Moreover, one can use "parallel transports" along integral curves of a vector field X to define a general covariant deriv- 3 ative of vector fields Y with respect to X. Considerable effort has been spent in the last fifty years to find intrinsic 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 is, an operator taking a vector field X on a manifold M and a section S of a vector bundle E over M to a further section VXS 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 parallel 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 all parallel families along o- . However, this is not how parallel transport arises in practice. Moreover, it is technically diffi• cult to formulate smoothness conditions in such a context. The way out is 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 of.all "initial 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 in the tangent bundle of the tangent bundle' So, one is forced to consider the "double tangent bundle" of a manifold. Actually, one gains in clarity by generalizing to the case of an arbitrary vector bundle. Accordingly, in Chapter Two we start by analyzing the structure of the tangent bundle TE of a vector bundle E over a manifold. The difficulties with the failure of the Cristoffel symbols to "transform like a tensor" show up again: TE is 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: it is 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 directly with geodesies by introducing the notion of a spray on a manifold Ms it is a vector field on the tangent bundle bundle of M whose integral curves "look like geodesies". In Chapter Four we recall the basic properties of sprays and show how one obtains the analogue of the Christoffel symbols. The main purpose of this exposition is 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 in a natural one-to-one correspondence. In Chapter Four we show how, on the tangent bundle of a manifold, sprays and "symmetric" connections are in 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 Kk. For an open subset U of M, let V(U) denote the vector space of smooth vector fields defined on U, and ^(E) the vector space of smooth sections of E over U. DEFINITIONS A covariant derivative on E is an operator V > T s V(M) X ^(E) M(E) ( X , S ) i > Vxs having the following properties: V S =V Dl) X+Y XS+VYS, fXs = fV D2) \7 xS, V V V D3) X(S+T) = XS + XT, and V fV D4) x(fT) = (Xf)T + xT, ( where f is 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 first 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) If Y e V(M) vanishes on U, then VyS vanishes .. on U for each S e ^(E), and (b) if T € fJj(E) vanishes on U, then \7XT vanishes on U for each X e V(M). PROOFt We will prove only (b), for (a) is even more straightforward. Fix some point y e U, and let f : M * IR be a smooth function, having support in U, such that f is identically 1 on some neighbourhood of y. Then by definition t fT « 0 € (E). Thus f 0 « VxfT = (Xf)T + VxT, so, at 7, we have 0 = (Xf)(y)T(y) + f(7)(VxT)(y) = (VxT)(y). Since y € U was arbitrarily chosen, the lemma is proved, PROPOSITION: If V is a covariant derivative on E, and U is 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), ( VvS)(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 is to define V on vector fields and sections which may not be extendable to global vector fields and sections. So, let X be in V(U) and S in H^(E), and fix some point y € U. Let f be' as in the proof of the lemma. Then f X e V(M) and fS € [^(E), so we may set (VjSKv) * ( VfxfS)(y). 9 The lemma ensures that V is well-defined, and by construction it is 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 let p ; E »M denote a smooth vector bundle over M with fibre Et . Let V be a covariant derivative on E, and let U be an open subset of M such that TM, the tangent bundle of M, and E are trivial 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 oU : C°°(U^Rn) X C~(U,IRk) > C°°(U,JRk). According to the definition in [1.1], this operator U S is linear over C^CUjJR) in the first argument but not 11 in the second. If, however, for F € C^Cu,© ) and G e C^CU^), we define TU(F,G) = SU(F,G) - DG(F) where DG : U > Lin(JRn,]Rk) is the derivative of G, then one checks easily that TU is linear over C°°(U,E) in both variables. 10 LEMMA: If an operator n k k T : C°°(U,IR ) X C°°(U0]R ) > C~(U,JR ) is bilinear over C^CUpIR), then (a) if F e G^U^IR11) vanishes at x « U, P(F,G) vanishes at x for each G e C°°(U,lRk), and (b) if G e C°°(U^Rk) vanishes at x € U, P(F,G) vanishes at x for each F e G°°(U,JRn). PROOF: We will prove only the first assertion. n For each i from 1 to n, let E^ € C°° (\J9B ) denote the constant function onto the i canonical basis vector ofJRn. 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) = iSf,(x 1 )r(E.,G)(x1 ) = 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$ if, 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 r^(dx,u) = U(F,G)(x), then, by the lemma, P. is well-defined; and, by con- struction, it is bilinear. Moreover, by definition the map TJ n k k p. > Bii(m x m , m ) Xl r > x is smooth. So? over U, V is represented by the smooth family k of bilinear P s mn X TR > in the sense that S U(F,G)(x) = DGl F(x) + T (F(x),G(x) ). |x x Consequently, V-^S(x) is already determined by the value of X at x and the values of S along any smooth curve fitting X(x). § 1.3s Parallel 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 cr , The covariant derivative of a section S of E along cr is the new section > VtS : I — E of E along cr defined as follows: For fixed t0 e I, we note that the map c* : ti »-(t6 + t) is a smooth curve representing cr '(t© ) = t| c ^cr(t Moreover, S is defined along 9 so we may form V ^S(t0 ), which we define to be the value of V^S at t0 . In local coordinates, if V is given by 1 n " . TO v^- 7D^ . ir»^ r • kl. UI. M.L , and S by G : I »IRk, one finds that S is given by , t, » G'(t) + r We say that a section of E along cr is parallel along cr if V^S E 0, Locally, finding sections parallel along cr means solving +£ (cr G'(t) (t) '(t),G(t) ) = 0 13 for G. Since this is a linear homogeneous differential 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) is a real vector space, the following (V,V) is well-definedo Let D denote the map D(V,V) : V(M) X^(E) >^ Clearly D(V,V) satisfies conditions Dl) to D3) of [1.1] and condition D4) yields: (V,V V V D )(X,fS) = x(fS)- x(fS) « (xf)s + fVxs - (xf)s - fVxs - f Vxs • f Vxs ,V) Thus the difference D(7 is bilinear over C~(MfR). V D(V,V) is called the difference tensor of and ^7 . Suppose now that V is any covariant derivative defined on E, and that T r D J V(M) X M(E) -> M(E) is bilinear (over C^M,©) ). Setting V « V - D, it is trivial to check that V is then a covariant 14 derivative of E, and that D(V,V) = D. Locally, over a suitable coordinate domain U in M9 we may represent V by a bilinear map Tu s C~(U,Bn) X C°°(U,Ek) > C°°(U,Rk), and V by a similar map TU. The difference D(V,vT is then represented by Tu' - which is 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 its tangent bundle. Recall that this is a smooth vector bundle over M with fibre IRn. A covariant derivative on M is a covariant derivative on TM as defined in [1.1], That is, an operator V s V(M) X V(M) > V(M) ( X , T )» > \7XY satisfying conditions Dl) to D4). Let U be an open subset of M such that rT1(U) is trivial. Then, as in [1.2.2], we may identify V(U) with C°°(UpIRn). Let X and I be smooth vector fields on U with corresponding maps F and G € C°°(tT,En) respectively,. Then, in 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 HRn is bilinear. Since V is determined locally by these P *s, we may derive an explicit representation of in terms of the coordinate system of U. § 1.5.2; Classical Notation We will continue with the notation of the last paragraph. Let E, denote the i canonical basis vector of JRn. Then we may write F = ,E. . n i i where each € C^CUjB), and similarly G = 2 g.E.. 3 3 3 Then, since Q is bilinear, we have I^CFUKGCX)) = Px( 2 fi(x)Ei, 2 gj(x)Ej) i j 2 f±(x)g.U) ^x(Ei,EJ.). i t J Moreover, each map rj\ : U >mn ^(E^E.) 16 must be smooth, so it may be written where each € C°°(U,m). Finally we have r(F(x),G(X)) = 2 ft(x)g.(x)[ 2 rt(x)E.], x 1, j 1 J .k 1J K so r , and hence V, is given locally by specifying n3 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 is, on the tangent bundle of M). Let cr : I ^ M "be a smooth curve. By a vector field along cr we mean a section to : I > TM of TM along cr. As in [1.3] we say that oo is parallel along cr if ^U> = 0. Note that the canonical lift of CT is a particular vector field or ' : I * TM along cr . We say that CT" is a geodesic with respect to V if G~ * is parallel along 0~~. So, in local coordinates, where V is represented by n n n Tx : E Xl >B , cr is a geodesic if and only if 0-"(t) +IJ(t)(crHt),cr»(t)) £ 0. In other words, to find geodesies means to solve an 17 explicit second order differential equation which is quadratic in 0*'. By the existence theory of ordinary differential equations, for each x e M, and each cr (0) = x, ando-'(O) = cj . Section 1.7: Difference and Torsion < « Let M be a smooth manifold of dimension n, and let V and V be covariant derivatives defined on M. As in [1.3It their difference tensor D : V(M) X V(M) > V(M) ( X , Y ) i > VXY ~ VXY is 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) VXX a VXX for each X € y-(M), and (3) S, the symmetric part of D, vanishes. PROOF; Clearly, (3) implies (2). Conversely, if 0 = D(X,X) = S(X,X) for all X e V(M), we obtain 0 = S(X+Y,X+Y) = 2S(X,Y) for all X,Y e V(M), since S is symmetric. Thus, (2) and (3) are equivalent. To see that (l) and (2) are equivalent, we work in local coordinates. By definition, a curve cr is a geo• desic for V if cr"(t) +rcr(t)( 0-'(t),cr'(t)) = 0, and a geodesic for V if cr"(t) +fjr(t)(crUt), cr'.(t)) = 0 (where T and T represent V and V ) • Since, for each x € M and ^ € TXM, there is a geodesic cr with tf(0) = y, and cr*(o) = <| we see that (1) means: [^(uju) = r^(u,u) for all u €IRn. Consequently, (l) is equivalent to (2). By the torsion tensor of a covariant derivative V on M, we mean the map Ty : V(M) X V(M) > V(M) defined by T7 (X,Y) = V2Y - VYX - [X,Y], 19 where [X,Y] is the usual bracket of smooth vector fields. It is easily verified that Tv is bilinear and alternating over C0 0 (M,JR). A covariant derivative V on M is said to be torsion-free if Ty s 0. Locally, if X and Y in V(M) are represented by F and G in C°° (-U,-En) respectively, then [X,Y] is represented by DG(F) - DF(G). Since V^Y is represented U by DG(F) +r (F,G), then we know Tv (X,Y) will be given by the map X 1 U » JRn xl >rx(G(x),F(x)) - Px(F(x),G(x)). Note that V is torsion-free if and only if each corres• ponding P. is 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 if A denotes the alternating part of the difference tensor of V and V , we have Tv - 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)TV . One may show by computation that V is the desired derivative. 21 Bundle Connections From the geometric viewpoint, it is desirable to characterize connections in terms of morphisms of vector bundles. A detailed digression on the vector bundles involved is required in 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 is a smooth manifold having a compat• ible group structure. That is, a structure under which multiplication and the taking of inverses are smooth Operations, we will show that the tangent bundle of a smooth group inherits a compatible group structure. The proof of this is greatly simplified if we express the definition of a smooth group in the language of diagrams. § 2.1.1s Definition of a Smooth Group DEFINITIONS A smooth group is 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): idQ X G X G X G -» G X G m X id G m m GX G -> G (2) there exists a smooth map *• of the one-point manifold into G such that the following diagram commutes (unit element): (idQ,*0 -> GX G id (*,IdG) G G X G -> G , and (3) the following diagram commutes (inverses): (idQ,i) * G X G m G. Note that all of the maps appearing in this defini• 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 its 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 is 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 MX N onto M, and pr2 the corresponding projection onto N. The definition of M X N ensures that these are smooth maps. Thus they will have smooth tangents: Tprx 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 is 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 pr2 ° ^ = P This induces a morphism of TM X TN into T(M X N) which is easily seen to be smooth. This new morphism is 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 is given locally by: Tm: ( (g,u), (h,v) )i >(m(g,h), T .m(u,v) ). If ©< 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 will be that vector in the fibre of TG over the unit element of G which represents the constant curve at that point. That is, 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• plicitly viewed under the decomposition to follow. Let m and i represent multiplication and inver• sion on G respectively, and let Tm and Ti be their associated tangent maps. From the definition of Tm as given in [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 is a smooth group homomorphism of TG onto G. Let OQ denote the canonical "zero-section" of TG. That is, 0Q : G >TG gi » (g,0). 26 Since 0 d T G is represented by the constant curve at S g, and tangent multiplication is essentially pointwise, 0^ is also a smooth group homomorphism. Moreover, p ° 0G is the identity map on G. This situation may be neatly described algebraically in terms of a semi-direct product. defined as follows. Let H and G be groups, and let 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 = id^. 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 this decomposi• tion, multiplication is given by: -i(k)sCg). KkOsCg') - i(k)[s(g)i(k')s(g-1)]s(g)s(g«) K is normal^ so, s(g)i(k *) s(g"*^") e i(K), and so the product is in the desired form. If we abuse notation, and let 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 is 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 is 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 let p and OQ be as defined in [2.1.3], We now identify ker(p) and the action of OQ mentioned in the last section. Clearly, as a manifold, ker(p) is just T (G). Note that, as a vector space, T G has the structure of an additive group, and that there is a canonical action of G on TeG 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, its tangent map at e, 28 ad(g) = Te(int(g)) : T@G >TQG must be linear. The resulting homomorphism ad s G >Lin(TeG,TeG) is knovm as the ad .joint representation. THEOREM; In the above notation; (a) the group structure on ker(p) induced by Tm is vector space addition in TgG, and (b) the action of G on ker(p) induced by the semi- direct decomposition of TG is the adjoint representation. PROOF: To see that (a) holds, we recall that the unit element of TG is the zero-vector in TeG. Therefore, since T m : T G X T G »T G e e e e is linear, Tem(^ , rr\ ) = TQm( ^ , 0) + Tem( 0, ^ ) = c* +^ . (b) follows from the fact that if ^ € TQG is represented by a curve crf then ad(g)(^j) is represented by the curve ti >gcr(t)g""1. Thus, if o t\— >^(t)cr(t)[oA(t)r1 But, by definition, this is the curve representing - ^(g^OoU"1), so we are done. 29 Note that ker(p), with this structure, is the additive group of the Lie-Algebra of G, denoted L(G). To sum up: if G is a smooth group, then TG is 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 is 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 if there exists a smooth map *t ..: . G X M ->M such that: (l) the following diagram commutes: m X idM G X G X M G X M idQ X G X M -» M, and (2) the following diagram also commutes: (*,idM) » G X M Clearly, if G acts smoothly on M, then TG will act 30 smoothly on TM with the tangent action. 2.1.6: Special Case; GL^ The smooth group that we will be interested in is GI»k, the group of all linear automorphisms of TR . k2 As an open subset of IR , GL^ has a smooth structure which clearly is compatible with the group structure. By definition, GL^ acts linearly (and hence smoothly) on lRk by: k k GLk X TR > lR (A , u) i *Au. As a manifold, the tangent bundle TGLV simply is GL^ X Mk, where M^ is the vector space of all (kxk)- matrices. To identify the group structure of TGL^, we look at the "tangent action" k k k k TGLk X IR X TR — >TR X TR , where we have identified TEk with 1Rk X Ek. Clearly the action is given by: (A,M,u,v)l >(Au,Mu + Av). Therefore, TGLfc can be identified with the subgroup of ^2k consisting of all matrices of the form fA 0" A e GI^jM € Mk M A 31 Note then that p is given by A 0* -» A, M A and OQ by A 0 A 0 A so the semi-direct decomposition is K A o" "I o' "A 0' • (MA"1^). 1 A_ _MA~ 1 .0 A. 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 is a smooth manifold, it has a tangent bundle which we denote nE : TE- »E. We call 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 is given by xi >-h(x) where h is a diffeomorphism between open subsets of 32 TRn, then the corresponding transition for the tangent bundle TM is given by: (x,dx)i > (h(x) ,h 9(x)dx). Then since local trivializations of E have transition functions of the form (x,u)i » (h(x),t(x)u), where t is a smooth mapping of an open subset of TRn into GL^, the corresponding transitions with respect to the E-structure of TE will 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 is 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 will 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 trivializa• tions of TE over TM in such a way that the transition functions act linearly on the fibres of Tp. To do this, fix some point in TE, and suppose that (U, <^ ,p) is a vector bundle chart at its. image under n-g : TE—>E. 33 Then since p i p°*1(U)^ -» cp(U) X TRk is a diffeomorphism, Tp s T(p~1(U)) >T(c£(U) X IRk) is also. But we know that the tangent functor T commutes with products, and from the definition of Tp 1 1 we get that TCp"" ^) ) = Tp" (TU)? so (TU,T 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 is a diffeomorphism of open subsets of TRn, and t is a smooth map from an open subset of TRn into GL^. If we restrict 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 is identical to the image of the tangent map of t at (x,dx). This lies in TGL^ which was estab• lished in [2.1.6] to be a subgroup of GL2k. So, the transition functions do act linearly on the 34 fibres of Tp, and we hare a vector bundle structure on TE. We call this the tangent structure of TE over TM, or simply the TM-structure. Note that, although the transition functions are linear in (u,du) and in (dx,du) separately, they are not linear in (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 all points (^,^) such that P(?) = ifc(j). Note that if p*" denotes the projection of E X TM onto E, M then we have a smooth vector bundle over E with fibre JRn, and if 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 locally by triples (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 in the first coordinate, so the image of (TT^T ) will be E X TM. E p M Denote this new morphism T( : TE >E X TM M As an immediate consequence of the definitions, we see that locally Tl s (x,u,dx,du) » :—>(x,u,dx). Tf has two interpretations. Considering TE and E X TM as vector bundles over E, TT is 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 is 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 in [2.3]. The following commutative diagram summarizes the relationships of the preceding section. E X TM Since Tt is both an E-morphism and a TM-morphism, it 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 is, we concentrate on the following part of the diagram: TT TE > E X TM M E PROPOSITION: There exists a smooth vector bundle morphism i : E X E TE M such that i EXE -» TE E X TM -> 0 M M is 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 fix some e € E. If p(e) = x e M, the fibres of TE and E X TM M over e are TQE and T^ respectively. So, by definition of Tf , the fibre of ?E over e will be the kernel of the tangent map of p at e. That is, «= kernel of TQp : TQE -* T^M In other words, V". is the tangent space at e to p™ (x) e —1/ E But p (x) = E is a vector space, so Y~* may be identi- fied with E . We now exhibit the formal proof of the proposition. PROOF: Define i : E X E > TE M locally 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 is 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 is, we concentrate on the diagram 3* TE » E X : TTMM M TM PROPOSITIONf There exists a vector bundle morphism j i TM X E TE M such that 0 * TM X E —^TE— * E X TM — > 0 M M is 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 first determine the fibre (TE)^ of TE over <| . By definition (TE)^ = Tp"1^). Thinking of TE as the disjoint union of its fibres over E we haves (TEL o U (TE)« PIT E = U ' Tp-1( 6 )'fl TE 5 e«E 5 e eeE 5 e 1 -But Tp"* ^) n TgE is the inverse image of ^ under Tgp. In particular, Tp^Cj) n TeE = p unless e is over xs that is, unless e is in the fibre Ex of E over x. 1 Suppose e is in Ex. Then Tp*" (t|) n TgE is the inverse image of under TQp and hence a translate of 39 the kernel of To, In [2.4.1] we identified the latter with E . Thus, as a set, U n U Ev = E E . (TEL = (TE)e TE = X e x . J eeE s e cB Note that, after this identification, 1f : TE * E X TM M acts on (TE)^ = Ex X Ex by projection onto the second coordinate. In other words, the fibre of the kernel of Tt over ^ may be identified with Ex. Now, the formal proof of the propositions PROOF % Define j s TM X E »TE M locally 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 is a smooth section C s EX TM >TE M 40 of TT which is 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 is an E^raorphism, f acts linearly in dx, and since C is a TM-morphism, f acts linearly in u, when the corresponding fibres are fixed. For each x e M then, C induces a bilinear map which we denote -T • IRnXTRkfc——>TRk by -P s (dx,u) « >f(x,u,dx). The negative sign appearing in the notation is intro• duced so as to ensure later agreement with classical notation. Locally then, a connection map is given by C : (x,u,dx)i > (x,u,dx, -UJ(u,dx)) where xi > -fZ is a smooth mapping of an open subset of TRn into the bilinear maps from TRnxmk to TRk. I 2.5.2; Connection Forms There is an equivalent formulation of [2.5.1] in terms of connection forms. Let M be a smooth manifold of dimension n, and 41 let 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 in [2.5.1], Recall the two short exact sequences of [2.4], That is9 i It E X E » TE— =>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 > EXE B M via i„ and Kmf : TE »TM X E M via j. These are given locally by Kg s (x,u,dx,du)> >(x,u,du +[^(u,dx) ) and Krpjyj : (XpU,dxydu)I »(x?dx?du +fx(u,dx) ) These agree in the third coordinate. Thus Kg and K^ induce the same map K s TE- »E given locally by K ;': (x;UpdX;du)i » (x,du + P(updx) ). Note that K has the following properties: 42 CF 1 s K is a vector bundle morphism over p : E—»M whose restriction to V is projection onto the second coordinate, and CF 2 : K is a vector bundle morphism over TT^ ? TM—>M TM whose restriction to V is projection onto the second coordinate, where and V™ denote the "E-kernel" and "TM-kernel" of TT respectively. DEFINITION; A connection form on E is 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 is 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 KC; and, alternatively, the connec• tion map corresponding to a connection form K by Cg.' Locally then, if C is given by C : (x,u,dx)i »(x,u,dx, -r(u,dx) ), then KQ is given by KC : (x,u,dx,du)» » (x,du +Tx(u,dx) ), and if K is given by K : (x,u,dx,du)* >(x,du +P(u,dx) ), then CK is given by CR : (x,u,dx)» »(x,u,dx, ~x(x,dx) ). 43 § 2„5.3: Alternative Definition of Connection Forms In Chapter Four we will be talking about geodesic sprays, and there a slightly different definition of connection forms will 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 locally 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 is a connection form on E if and only if CF 1 is satisfied, and: CF 3 : K os * «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 is a smooth map satisfying CF 1 and CF 3. Then, as we have seen, CF 1 implies that locally K has the form K : (x,u,dx,du)l »(x,du + [^(ujdx) ) where P^Ujdx) is linear in dx. By CF 3> P^sujdx) = sl^Ujdx), so, P„(u,dx) is also linear in u (cf Appendix). Thus CF 2 of [2o5.2j is satisfied, so K is a connection form on E. The converse is also easily established, so the proposition stands. Section 2.6; The Difference Field of Two Bundle Connections The two characterizations of bundle connections in [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 in [2.5.1] and suppose they are given locally 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 -is given locally by Dg(C,C) ; (x,u,dx)r > (x9VLfOt-Vx{\ifdx.) + I"x(u,dx) ). 45 E The image of DE(C,8') thus lies in V , the E-kernel of ff , so under the identification of [2,2.4], we arrive at an induced morphism given locally by (xsu,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 locally 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 call the difference field of C and cf. Note that the linearity requirements on ' and V ensure D(C,C) acts linearly fibrewise. Suppose now that C is any connection map on E, and that D : E X TM >E M is a morphism as above. Then if C is given locally by C s (x,u,dx) i -> (x,u,dx,-r_(u,dx) ), and D by D : (x,u,dx)i > (x,fx(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 is easily verified that C is well-defined, and is 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 locally 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 is 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 identifica• tions of [2.5.2] we have an induced morphism given loc• ally by (x,u,dx)t ± (x,rv(u,dx)-r.(u,dx)) where (x0u9dx) e E X TM. M Similarly, seen as vector bundle morphisms over TT^ t TM *M the difference Bm(K$) annihilates TM V . This induces a morphism given locally by (x,u,dx) \ »(x, rc(u,dx)-r?(u,dx)). That is, under our identifications, Dg(K,ft;) and Drj^jCKjK*) induce the same morphism. We call this the difference field of K and f, and denote it simply D(K9%). Finally, note that if 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 latter is as defined in [2e6.1]. The minus sign appearing in the formula is the result of the earlier decision to define the P. 's to agree with more classical 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 is of the form S : xi >(x,u(x) ) where u is a smooth mapping of an open subset of Bn into E . It follows that S^ in the corresponding coordinate domains will be given by s (x,dx)\ > (x,u(x) ,dx,u *(x)dx). The follovring property of such morphisms is trivial PI s (S + T)^ = S^ + T^ There is, 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 is another smooth section. Locally, if S is given by S 8 xi > (x,u(x) ) then fS is given by fS s xi > (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 first 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 is i : TM X E > TE M . (x,dx,u)» >(x,0,dx,u). The second term of the decomposition lies in 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,fv (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 is the product of df : TM > IR with S. 50 This establishes the following property? -8- P2 : (fS)* = fSn i(dfS) in 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 in [3»l.l]. Then if X and Y are smooth vector fields on M, and f is a smooth real- valued function defined on M, the following hold: i) S* -(X + Y) -• S^.X + S*.Y, (fx) ii) S* o = fS*«>X, x iii) (S + T)* o = ° X + T^° X. and iv) (fS)^o X «' fS* * X 4- i(XfS) where i is the injection defined in [2.4.2]. The proof of the properties is based on the fact that the following two diagrams commute: TE 51 TT where ^ and pE are the vector bundle structural maps of TE over TM and E respectively, and df is the second component of Tf : TM —TJR ^ffiXE. 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 in [2a5«2]„ For a smooth vector field X on M, and a smooth section S of p, define VXS « K ° S* • X where S# = TS« That is, the following diagram commutes: M By conditions CF 1 and CF 2 of [2.5v2].f and i) to ivj of [3»1»2], V is a well-defined covariant derivative on E, Note that over a fixed coordinate domain in M, suit• able choices for charts give the following formulas: K s (x,u,dx,du)i »(x,du + HL(u,dx) ), X : xi > (x, | (x)), and S : xi >(xfu(x)): so S^»X : xi > (x,u(x), t| (x) ,u'(x)|(x)), and VXS : x i » (x,u •(*) <§ (x) + i^(u(x), Section 3»2; From Covariant Derivatives to Connection Forms ~ In [3«1] we established that connection forms induce covariant derivatives. We now show that one may go the other way. I 3«-2.1; Uniqueness Let M be a smooth manifold of dimension n, and let p : E »M denote a smooth vector bundle over M with f ibre IR . Fix e e E, and suppose p(e) = x e M. Let S be a smooth section of p such that S(x) = e. Then, in the notation of [3»1]» for any smooth vector field X on M we have (S*°X)(x) € TeE. Let UV denote the set of elements of T^E of this form, e e That is, u = x)(x s x e x e ° ) ( > = » arbitrary} . Recall, from [2.4»1]» that the double morphism Tt : TE v E X TM M E is an E-morphism having an E-kernel V . We claim that U U vf = T^E. e e e In local coordinates, points of TE are of the form (x,u,dx,du)| and, fpr e = (xo,u0), vf is given by: 'vj = {(x ,u ,0,du) du € Ek } . So, we must establish that U*e contains all points of the 53 form (x0,u0 ,dx,du) with dx ^ Oo Note that, for u0 and k n du in IR , and dx / 0 inK , we can find a smooth map F : TJ >Ek, where TJ is some neighbourhood of x , such that F(x0) = u, and F'(x0)dx = du. Now, there exists a vector field X on M with X(xe) = dx, and a section S of E which agrees with F on a neighbour• hood of x0, so we are done. APPLICATION; If 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 VE, 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 fibre Ek. Let V be a covariant derivative on sections of E. We wish to define a connection form on E inducing V as in [3-1]. By [3.2.1] it suffices to find K locally. Recall that in local coordinates we may write: V (x) + T XS s xi *(x,u'(x) c5 x(u(x), t| (x))) 54 where -f : IRk X Htn IRk is bilinear and depends smoothly on x. We define K locally by K s (x,u,dx,du) (x,du + P(u..dx) ). It is easily verified that the global -morphism K. so. defined satisfied the conditions of a connection form. Section 3.3: Differences Note, in the foregoing notation, that if K and K* are connection forms on E inducing covariant derivatives V and V respectively, then the difference field defined in [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 rK(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 recall that it is a smooth vector bundle over M with fibre 3Rn. Since TM is a smooth manifold of dimension n2, it has a tangent bundle, called the double tangent bundle of M, which we denote by TTfyj : TTM —-. > TM. Recall that if a point of M lies within the domain of a chart (U, At the tangent level, points of the corresponding domain are denoted by pairs (x,dx) e <^(U) X 3Rn, and the co• ordinate change is given by the diffeomorphism —> (x,dx) 1 ; > (h(x),h'(x)dx). Since TTM is the tangent bundle of TM, we will 56 denote points by quadruples (x,dx,x,dx) € (y(U)> CD(W) X ffin XlnXffin > ^(W) X EN X JRN X EN v (xodXyXodx) t > (h(x),h (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 is a vector bundle over M, TTM will have the double structure discussed in [2.2], In particular, nm : TTM > TM gives the standard structure of TTM, and we have the following exact sequence of smooth vector bundles over TMs i 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 is given by: TT^ : TTM >TM, and another exact sequence over TM is given by: 57 •4 TM X TM -> TTM -> TM X TM » 0, M M where, again in local coordinates, Tr^j c (x,dx,x,dx) i > (x,x), and j : (x,dx,x)i >(x,0,x,dx). It is important to note that these two structures are Isomorphic That is, If I^r^ denotes the involution of TTM given locally by )i 'TTM s (x,dx,x,dx *(x,x,dx,dx), and ITM denotes the canonical involution of TM X TM, TM M then the following diagram commutes: i -* TM X TM -•TTM —> TM X TM M M ->0 •TM I TTM ITM -> TM X TM •> TTM •^TM X TM M 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 in [2o5'.l]. That is, a smooth section C : TM X TM » TTM M of Tf (as defined in [4<>1»2]) which is a vector bundle morphism with respect to both the standard and tangent 58 structures of TTM. Similarly, a connection form on M is defined to be a connection form on TM. That is, a smooth morphism K s TTM. —-. TM satisfying% CF 1 s K is a vector bundle morphism whose restriction to i(TM X TM) is projection onto the second M coordinate, and CF 2 : K is a vector bundle morphism over whose res• triction to j(TMX TM) is also projection onto M the second coordinate, where i and j are as in [4e>1.2], In [2,5,3], we saw that condition CF 2 is 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] it was shown that connection maps and connec• tion forms are in a one-to-one correspondence. Recall that locally, connection maps have the form 59 ! C s (x,dx,x)i > (xydXyXp-^CdXjx) ) where Q is a bilinear mapping of JRn X lRn onto Bn depending smoothly on x. Corresponding to each such C we have a connection form K^, given locally by ^C KQ S (xjdXyXpdx) t (x,dx + P dx,x) ). Since the two bundle structures of TTM are isomorphic, we have a notion of symmetry. That is, we say that a connection map C on M is symmetric if it commutes with the involutions of TTM and TM X TM introduced in [4.1-2]. That is, when c"'° % = ITTMeCo Correspondingly, a connection form K on M is symmetric when K ° TTM = K* In local coordinates, this means K and C are symmetric if and only if each of the corresponding P *s is symmet- ric. Thus C is symmetric if and only if is symmetric. Finally, recall that for any two connection maps C and C on M there is a morphism D(C,C) : TM X TM >TM M called the difference field of C and C, and that given any 60 connection map C, and a morphism D as stated,, there is a unique connection map C on M such that D(C,C) = D. No.te that if C and C are symmetric, then so- is 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 its tangent bundle, and TT^ S TTM >TM its 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 is an isomorphism of these two structures. A second order differential equation on M is defined to be a smooth vector field X : TM » TTM which is a section of both the standard and tangent structures of TTM. That is, such that nTMoX = idTM ° TnM°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 ^ is a smooth IRn~valued map. By definition, a smooth curve p : I > TM is an integral curve of a vector field Y on TM if and only if p'(t) = Y(^(t)) for each t in I, where p ' ; I > TTM is the canonical lift of /3 to the tangent bundle of TM. In local coordinates, p may be written ti > (cr (t),t (t)) €UXB^ Therefore, if X is a second order differential equation with local representation (x,dx)i » (x,dx,dx, ^(x,dx)), p is an integral curve of X if and only ifs cr'U) = t(t), and t'(t) = In other words, the condition is that X = CT', and cr " «|(cr,cr'). Therefore, integral curves of second order differential equations are equal to the canonical lifts 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 in [4«2.2], If we restrict C to the diagonal of TM X TM, then M the resulting map induces a second order differential equation . XQ % TM —> TTM called the spray of C. In local coordinates, if C is given by C : (x,dx,x)i »(x,dx,x,- (dx,x)) then is given by XQ : (x,dx)i > (xjdx^dXy-r^CdXydx)). We may characterize those second order differential equatxons which arise from ccnnectxon maps as fellows! For s € R, let s : TM > TM and s s TTM » TTM denote fibrewise multiplication by s, and let s^. denote the tangent map of s0 Then if X^ is the spray of a connection map C on M, the following condition holds: ° (Sp) s*<> s * XC = XC s, since each Hy. is quadratic. In general, a smooth morphism X : TM * TTM is called a spray on M whenever it is a second order differential equation on M satisfying condition (Sp). Locally then, if X is given by X % (x,dx)l > (x,dx,dx, <^(x,sdx) = s2^(x,dx), so a second order differential equation is a spray if and only if | is a quadratic form in 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 in the last section. THEOREM; There is a unique symmetric connection map C ; TM X TM > TTM M on M generating the spray X. PROOF; Suppose first that C and C are two symmetric connection maps generating the same spray X. Consider their difference as in [2.6.1]. By hypothesis, it will vanish on the diagonal of TM X TM, and so must be alter- M nating. On the other hand, it is symmetric. Therefore, A/ it must vanish everywhere. Thus C = C. Since uniqueness is established, we need only exhibit a suitable map C locally. Suppose X is given locally by X : (x,dx)t > (x,dx,dx, <|x(dx)) where jpx(dx) «= ^(x,dx) is quadratic in dx. Since^x is homogeneous of degree two, it is the restriction to the diagonal of some bilinear mapping of 3Rn X IRn into lRn (cf Appendix). The family of all such maps inducing |x 64 contains a unique symmetric member. Denote this map by -To The following is 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 field T, called the torsion field of C. Thus, although sprays are not in one-to-one corres• pondence with bundle connections or covariant derivatives, the explicit formulas in 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 is torsion-free if and only if K and C are symmetric, (2) a smooth curve on M is a geodesic of V if and only if it is the projection to M of an integral curve of X, and (3) the torsion tensor of V is equal to the torsion field of C. 65 BIBLIOGRAPHY 1. W. AMBROSE, R. S„ PALAIS, I, M. SINGER^ "Sprays", Anais du Academia Brasileira de Ciencias, vol. 32, 163-173 (I960) 2E J. DIEUDONNE: "Elements d'Analyse", tome III, Gauthier-Villars, Paris, (1970) 3. Co GODBILLONs "Geometric Differentielle et Mecanique Analytique", Hermann, Paris, (1969) 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; (1962) 66 APPENDIX Homogeneous Smooth Functions Recall that a function f s &n > B*1 is homogeneous of degree k ^ 0, if and only if, for each u € Bn, (H) f(su) skf(u) for any s e B. Note that if f is homogeneous of degree 0, then (H) implies that f is a constant function, and if f is homogeneous of degree k > 0, then f(0) =0. LEMMA: If f is a differentiable homogeneous function of degree k > 0, then Df : Bn > Lin(Bn, B01) is homogeneous of degree k-1. PROOF: Differentiating (H) with respect to u gives; sDfl = skDf| , l(su) lu so DfI = sk_1DfI l(su) l(u) LEMMA: (Euler's Relation) If f is a homogeneous differentiable function of degree k > 0, then Df| (u) = kf(u). lu PROOF: Differentiating (H) with respect to s gives Df I (u) = ks^fCu). l(su) But by the above lemma. Df| (u) o s^Dtl~ f (u), (su) 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, PROPOSITION: A smooth function F S JRH ^m is homogeneous of degree k > 0 if and only if there exists a k-linear function f : IRn X Bn X ..... X ]Rn ?TRm such that for each u € IR11 , £(u,u, .....,u) =s f(u).