Modern differential geometry for physicists

MODERN DIFFERENTIAL GEOMETRY For Physicists

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World Scientific Lecture Notes in Physics Vol. 32

MODERN DIFFERENTIAL GEOMETRY For Physicists Chris J Isham Theoretical Physics Department Imperial College of Science and Technology United Kingdom

VQ World Scientific Hi

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd., P O Box 128, Fairer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging-in-Publication Data Isham, C. J. Modern differential geometry for physicists/Chris J. Isham. p. cm. — (World Scientific lecture notes in physics; vol. 32) ISBN 9971-50-956-3 ISBN 9971-50-957-1 (pbk) 1. Geometry, Differential. I. Title. II. Series QA641.184 1989 516.3'6--dc20 89-16487 CIP

Copyright © 1989 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any forms or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

These notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by our first-year theoretical physics PhD students, or by students attending the one-year MSc course "Fundamental Fields and Forces" at Imperial College. The geometry course is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen with an eye to the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma-models and other types of nonlinear field systems that feature in modern quantum field theory. The course is in three parts dealing with respectively (i) introductory coordinate-free differential geometry, (ii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds, (iii) introduction to the theory of fibre bundles. In preparing the notes I was particularly conscious of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry. In particular, in the first part of the course I have laid considerable stress on the basic ideas of "tangent space structure"

which I develop from several different points of view: some geometrical, and others more algebraic. My experience in teaching this subject for a number of years is that a firm understanding of the various ways of understanding tangent spaces is the key to attaining a grasp of differential geometry that goes beyond what is, all too often, merely an acquiescence in the jargon of the field. Part two of the course is concerned with the theory of Lie groups and the action of Lie groups on differentiable manifolds. I have tried to emphasise here the geometrical foundations of the connection between Lie groups and Lie algebras, but the latter subject is not treated in any detail and readers not familiar with this topic should supplement the notes at this point. The final part of the course contains a short introduction to the theory of fibre bundles and their use in Yang-Mills theory. This is fairly brief since many excellent introductions to this subject aimed at physicists have been published in recent years and there seemed no great point in replicating the material in detail here. Finally, let me emphasise that what is presented here are precisely the notes as handed out by me to the students to avoid having to write too much on the blackboard. In the context of the course, this written material is naturally reinforced by my verbal comments. I must apologise if the reader finds that the absence of these expository remarks renders some of the material rather pithy in places.

VI

CONTENTS Preface

v

1. Differentiable Manifolds 1.1 Main Definitions 1.2 Tangent Spaces 1.3 Tangent Vectors as Derivations 1.4 Vector Fields 1.5 Integral Curves and Flows 1.6 Cotangent Vectors 1.7 General Tensors and n-Forms 1.8 DeRham Cohomology

1 1 6 12 24 31 40 46 53

2. Lie Groups 2.1 Basic Definitions 2.2 The Lie Algebra of a Lie Group 2.3 Left-Invariant Forms 2.4 Transformation Groups 2.5 Infinitesimal Transformations

61 61 67 82 87 101

3. Fibre Bundles 3.1 Bundles 3.2 Principal Fibre Bundles 3.3 Associated Bundles 3.4 Vector Bundles 3.5 Connections in a Principal Bundle 3.6 Parallel Transport

Ill Ill 123 135 151 154 164

Index

179 vii

1 DIFFERENTIABLE MANIFOLDS

1.1. MAIN DEFINITIONS Definition A (finite dimensional) differentiable manifold is a (connected) topological space with the properties: (i) M is locally homeomorphic to IRm for some m < oo. Thus, around any point/? e Mthere exists an open neighbourhood £/and a homeomorphism / from U into an open ball in Um.

(ii) If (Uu 4>\) and (U2, 2) are two such coordinate charts then the overlap (or transition) functions 2°\~1 '• 4>\{UlC\U2) -* 02 (Ui n U2) are smooth (i.e., infinitely differentiable, or C00)

l

2

Modern Differential Geometry For Physicists

Comments (a) A point peUcM, has the coordinates (4>\p), m m 4>\p),..., 4> {p)) e U with respect to the chart (U, ) where the functions ': £/-»• IR, / = 1 , . . . , m, are the coordinate functions defined as ' (p): = u'{4> (p)) in terms of the slot function u': Um-+ U, u'{x): = x'. The coordinate functions are often written as x', i = 1, ..., m, and the coordinates of a particular point p as (x1 (p), x 2 (/?), (b) There are analogous definitions of Ck (k) and (V, \[/) on M and N respectively, is the map ^o/o^-' : (U) c um

•W .

(b) A function f:M -* N is a C-function if, for all coverings of M and TV by coordinate neighbourhoods, all its local representatives are C r functions. [Note. Fortunately, it suffices to show this for one system of coordinate charts for each manifold. It then follows automatically for all other coordinate systems. Exercise!]

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Modern Differential Geometry For Physicists

(c) A function/:M—>- ./Vis a C-diffeomorphism iff is a one-to-one, surjective map such that both / a n d its inverse are C r functions. Note, (i) Two manifolds that are diffeomorphic (i.e., there exists a diffeomorphism between them) can be regarded as being "equivalent" in an appropriate sense, i.e., they are concrete copies of a single abstract Platonic manifold situated somewhere in the realms above. This is in the same sense that two groups that are isomorphic are regarded as being the "same" group. (ii) The set of all diffeomorphism of a manifold M onto itself forms a group Diff(M). Groups of this type play an important role in several branches of modern theoretical physics. 1.2. TANGENT SPACES One of the most important ideas in differential geometry is that of a "tangent space" to a manifold. This is based in part on the intuitive geometric idea of a tangent plane to a surface: TxSn := {velir + 1 |x-v = o} is the tangent space to the sphere S" at the point xeR"+\ but tangent space structure is also deeply connected with the local differentiable properties of functions on the manifold and this gives a more algebraic slant to the idea. There are in fact several, superficially quite different, definitions of a tangent space, some of which emphasise the geometric picture and some the algebraic. For finite-dimensional manifolds it can be shown that these definitions are actually equivalent and it is largely a matter of taste (and the details of the problem in hand) which determines the approach to be adopted in any particular case. The situation with

Differentiable Manifolds

1

regard to infinite-dimensional manifolds is however somewhat different and many of the finite-dimensional definitions are no longer appropriate. One of the main aims of this part of the Course is to present these different approaches to the definition of a tangent space and to show that they are equivalent. Thus we are firmly in the world of finite-dimensional differential geometry but I will start the presentation with the definition that does generalise easily to the infinite-dimensional case. This particular approach is slanted heavily towards the geometric view of a tangent space as being "tangent" to the manifold, rather as in the picture above for the case of a sphere. The problem with this picture however is that it is just that — a picture. In particular, it involves embedding the sphere in a particular euclidean space 1R"+1 and then regarding the tangent space as a specific linear subspace of this vector space. However, one of the aims of modern differential geometry is to present ideas in a way that is intrinsic to the manifold itself — not dependent on an embedding in some (largely arbitrary) larger dimensional vector space. In particular, the basic definition of a differentiable manifold made no reference to such a space, and neither should the definition of a tangent space. The critical question is therefore "What should replace the intuitive idea of a tangent vector as being tangent to a surface in the usual sense and "sticking out" into the containing space?" The answer lies in the idea of a tangent vector being tangent to a curve in the manifold. In the figure above, it is clear that a number of curves can be drawn on the sphere with the property that, at the point xeS", their tangent vectors (in the usual sense of curves in euclidean space) are equal to the vector v. The crucial point here is that the curves themselves lie in the manifold S", not in the surrounding IR"+' and that therefore this idea of a tangent vector as being defined in some way as 'tangent to a curve' can be generalised to an arbitrary manifold without embedding it in a

Modern Differential Geometry For Physicists

8

higher dimensional vector space. The general idea therefore is to actually define a tangent vector in terms of a curve to which it might pictorially be deemed to be tangent. However, things cannot be quite as simple as this because, for example, it is clear that in the case above of S" which is embedded in a convenient space, there are many curves on the manifold that are 'tangent' to the given vector v:

This leads to the idea of a tangent vector as being an appropriate equivalence class of curves, and this is the definition that we shall now present. Definition (a) A curve on a manifold M is a smooth (i.e., C°°) map a from some open interval (—£,£) of the real line into M. [Note. The "curve" is defined to be the map itself, not the set of image points in M. It is frequently important to remember this distinction.]

(b) Two curves ox and o2 are tangent at a point p in M if (i) (7,(0) = ff2(0) = p

(ii) In some local coordinate system (xl, x2,..., xm) around the point the two curves are 'tangent' in the usual sense as curves in IRm

Differentiable Manifolds

dx'

~dl

(ffi(0) r=0

dx1 (o (t)) ~dt 2

9

fori = l,...,m.

(1.2.1)

(=0

Note. If ax andCT2are tangent in one coordinate system then they are tangent in any other coordinate system covering the point peM. Thus the definition is an intrinsic (i.e., coordinate system independent) one. (c) A tangent vector at p e M is an equivalence class of curves in M where the equivalence relation is that two curves shall be tangent at the point p. We write the equivalence class of a particular curve a as [a]. (d) The tangent space Tp M to M at the point p is the set of all tangent vectors at the point p. The tangent bundle TM is defined as TM := [J TPM. There is a pe M

natural projection map n : TM -* M that associates with each tangent vector the point p in M at which it is tangent. The inverse image (i.e., the "fibre") of any point p under the map n is thus just the set of all vectors that are tangent to the manifold there, Tp M. This is a very special case of the general idea of a fibre bundle to which we shall return later. Comments (a) This definition of a tangent vector is consistent with the intuitive geometrical one when, as in the case of a sphere embedded in a euclidean space, there is a natural way of representing tangent vectors as elements of the containing vector space. This also extends to the tangent bundle TM which, in the case of the sphere, can be represented as:

Modern Differential Geometry For Physicists

10

TSn = {(x,v)elR''+1 X IR" +1 |x-x = 1 andx-v = o}. (b) A tangent vector v in TPM can be used as a "directional derivative" on functions/on A/by defining:

v(f) := -f{o

(1-2.2)

it)) (=0

where a is any curve in the equivalence class represented by v, i.e., v = [a].

[Exercise: Show that this definition does not depend on the particular choice of a in the equivalence class v.] The idea that a tangent vector can be regarded as a type of differential operator is a crucial one which lies at the heart of the other, more algebraic, definitions of tangent space structure. We shall return to this point later. In the heuristic, pictorial, representation of tangent vectors to S" embedded in IR"+I, it is clear that any two tangent vectors at a point p can be added to give a third, and that a real multiple of a tangent vector is itself a tangent vector. This suggests strongly that it should be possible to make TPM into a vector space in the general definition given above. Theorem

The tangent space TPM carries a structure of a real vector space.

Proof

In the diagram above, c, and a2 are two representative curves

Differentiable Manifolds

11

for the two tangent vectors i>, and v2 in TpM. We use a coordinate chart (U, 4>) aroundp e Mwith the property that <j>(p) = 0 e IRm and then consider the image curves 4>°al and °er2 which map an open interval into Um. We cannot of course "add" the curves cr, and c2 directly since the set M in which they take their values is not a vector space. However, the local space Um is a vector space and hence we can legitimately consider the sum t-~~~°o{(t) + 4)°o2{t) which is a curve in Um and which, like both er, and o2, passes through the null vector 0 when t = 0. It follows that the map t~~-4>~l°(4>°ox (t) + 4>°o2(t)) is a curve in M which passes through the point p when t=0. We then define: (i)

u, + u2 := [(A"'o (0off, + 0(72)]

(1.2.3)

and similarly, if v = [a], (ii)

rv :=["' o (rtfoo)]

for

all r in R.

(1-2.4)

It can be shown that [Exercise!] (i) These definitions are independent of the choice of chart (U, ) and representatives a{ and er2 for the tangent vectors Vi and v2. (ii) They make T^M into a real vector space. Note. This means that TM is a vector bundle. More details of this later. A tangent space can be regarded to some extent as a local "linearization" of the manifold and it is a fact of considerable importance that a map h between two manifolds M and TV can also be "linearized" with the aid of the tangent spaces and the vector space structure that they carry. This is the concept of the "pushforward" map h* that maps the tangent spaces of M linearly into the tangent spaces of N:

12

Modern Differential Geometry For Physicists

Definition If h : M -* N and v e TPM then we define h*(v) in Th{p) TV by h*(v) := [h°a]

where v = [a].

(1.2.5)

Exercise (a) Show that this definition is independent of the particular choice of the curve a in the equivalence class v e TPM. (b) Show that h* is a linear map, i.e., for all vu v2 e Tp Mand r e U, h* (t^ + t^) = h* (u,) + h* (v2)

(1.2.6)

h*(rv) = rh*(v).

(1.2.7)

(c) If M, Nand Pare three manifolds and if h:M-+ Nand k:N—*P then show that (k o h)* = k* h*.

(1.2.8)

1.3. TANGENT VECTORS AS DERIVATIONS We recall Eq. (1.2.2):

v{f) := JtfW))

where [o] = v

(1.3.1)

1=0

which enabled the equivalence class of curves v e TPM to act as a type of differential operator on the space C°°(Af) of real-valued differentiable functions on M. The first "algebraic" way of

Differentiable Manifolds

13

defining tangent vectors is to turn Eq. (1.3.1) around and define TPM as a space of "directional derivatives". This is developed by thinking of Cco(M) as a ring over IR, and of a tangent vector as a derivation map from this ring into IR: Definition (a) A derivation at a point p e M i s a map v:Cx{M) -*• IR such that (i) v(f+g) = «(/) + v(g) for a l l / g e C°°(M) »(r/) = rv(f) for a l l / e C°°(M) and r e U

(1.3.2) (1.3.3)

(ii) v(fg) = ftp) v(g) + g(p) v(f) for a l l / # in C»(M).(1.3.4) (b) The set of all derivations at /> e M is denoted Z)pAf. Comments (a) Eqs. (1.3.2-3) merely assert that v is a linear map from the vector space CX(M) to IR. The term "derivation" is used because of the crucial "Leibniz" rule in Eq. (1.3.4) which is clearly analogous to the rule in elementary calculus for taking the derivative at a fixed point of the product of two functions. (b) The space of derivations DPM can be readily given the structure of a real vector space by defining: (wi + « 2 ) ( / ) : = « i ( / ) + f2(/)

(1-3-5)

v(rf):=rv(f)

(1.3.6)

for a l l / „ / 2 , / in Cm{M) and r e U. (c) Eq. (1.3.1) defines a map fi: TPM-»• DPMwhich is clearly linear and injective (i.e., one-to-one). We will show below that it is

14

Modern Differential Geometry For Physicists

also surjective and is therefore an isomorphism. It is this fact that enables us to view tangent vectors geometrically as equivalence classes of curves, or algebraically as derivations of CX(M). A very important example of a derivation is provided with the aid of any coordinate chart (U, ) containing the point p of interest. Define

du"

f°