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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[^(<r(t)) oj'(t) oj'(t) = o where the \^ . are the classical Christoffel symbols. 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. <r"(t) + Q-(t)( cr'U), cr»(t)) « Of course, the V\ 's depend on the local coordinates one uses. The basic difficulty encountered by the 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<j- with the tangent space of M at cr(t0). Put differently, parallel families give a specific way of propagating tangent vectors along curves. 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.
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