Lectures on Fibre Bundles and Differential Geometry

138 Pages·1986·7.729 MB·English

by  J. L. Koszul (auth.)

Lectures on
FIBRE BUNDLES AND DIFFERENTIAL GEOMETRY
By
]. L. KOSZUL
Notes by
S. RAMANAN
Published for the
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
i
Springer-Verlag Berlin Heidelberg GmbH
Author
J. L. KOSZUL
Mathcmatiqucs
Universite de Grenoble
BP 116, 38402 Saint Martin d'Heres
FRANCE
CC') Springer-V crlag Berlin Heidelberg 1986
Originally published by Tata Institute of Fundamental Research 1986.
ISBN 978-3-540-12876-2 ISBN 978-3-662-02503-1 (eBook)
DOl 10.1007/978-3-662-02503-1
No part of this book may be reproduced in any form
by print, microfilm or any other means without written
permission from the Tara Institute of Fundamental
Research, Colaba, Bombay 400 005
CONTENTS
Page
Introduction
·. .
I Differential calculus
·. . ·. .
II Differentiable bundles 28
·. .
III Connections on principal bundles 41>
.. ·. .
·
IV Holonomy groups 63
·. .
V Vector bunPles and derivation laW8 77
·. . ...
VI Holomorphic connections 105
·. .
References 125
INTRODUCTION
The main topic of these notes is the theory of connections.
There are two basic notions in the theory: the notion of covariant
derivation which concerns differentiable sections of vector bundles,
and notion of connection forms on principal bundles. These two
th~
notions Hre by no means independent of each other. While any law of
covariant derivation in a vector bundle can be defined by a connection
forn in the princip21 bundle of framee, an independent treatment of
covariant is desirable in view of many applications
doriv~tions wh~re
the principal bundle remains in the background.
In the first chapter, we start with an algebraic formulation
of covariant derivations. The rela"i:,ed notions of curvature, differen
tials and torsion are discussed without reference to manifolds.
Chapters II, III and IV are devoted to a study of connection forms on
principal bundles. Chapter V deals essentially with the relations bet
ween covariant derivations and co:mcction forms. Some special features
of the theory of connections in almost complex and holomorphic bundles,
which include the recent results of Atiyah ~2-1 form the subject
matter of the final chapter.
We have not dealt with any topic related to the theory of
characteristic forms or that of "Cartan connections" in the sense of
11
Ehresmann, but a few referencos in these directions are given in the
bibliography.
An explanation of tho notations used is given at tho
beginning of tho notes.
Some notatjons n~ed in the notes
Let V be a differentiable manifold. A vector \'lith origin
at a point x E. V is usually denotod by dx. For every differen
vi
tiable map f of V into a manifold ,f(dx) is the image of dx
under the map derived from f; thus f(dx) is a vector \'lith origin
at f(x). Let G bo a Lie group. If (x, s) is a point of V ~G,
then (dx, s) is the ~..m3.ge of tho vector dx of V under the map
s ---7 (z, s) of V into V 1-- G. Similarly, (x,ds) is the image
of the vector ds tmd'3r t.he map t ---7 (x, t) of G into V >< G.
Both are vedors of V 1- G with origin nt (x, s). vie define (dx,ds)
to be (dx,s) + (x,ds). If G acts differentiably to the right in
V and if the r::.s.p V 'f.. G ---::;,. V is denoted as a. product, then dx s
and x ds are respectively the :i..m.3.ges of (dx, s) and (x,ds) under
the m<lp (z,t) -~ % t.
If L is a vector space, the v~ctorG of the I!lC.nifold L
with origin at 0 ;:,.re irkntifind ill the: ncltural ¥D.y with elements of
L. Thus, if f is a differcntiabl~ function on V with values in
L, dx ~ f(dx)-f(x) is a form of degr~ 1 on V with values in
L which is called. 'Lhe difi'erential of f and denoted by df. Let
I be an interval in Rand dl t be the unit vector of R with
origin at t ~ R. 'rhen for any differentiable map f of I into
a manifold V and e7ery tEl , f' (t) is the vector with origin
at f( t) image of d t und-er 1'0
l
Chapter I
Differential Calculus
1• 1
Let k be a commutative ring with unit and A a commutative
and associative 3.lgebra over k having 1 as its unit element. In appli-
cations, k will usually be the real number field and A the algebra
of'. cl.ifferE.Vltiable functions on a manifold.
Definition 1. A derivation X is a map X A ~ A such that
i) X E Ho~ (A ,A), 3.nd
11) X (ab) = (Xa) b + a (Xb) for every a,b EA.
If no non-zero element in k an:1ihilato3 A , k can be
identified with a subalgebra of A and with this identification we
have Xx = 0 for every .x E..k. In fact, we have only to take
a=b=l in (ii) to get Xl = 0 and consequently Xx = xX(1) - 0 •
\<le shall denote the set of derivations by C. Then C is
obviously an A-module with the following operations:
(X + Y) (a) = Xa + Ya
(aX) (b) = a(Xb) for a,bEA and X,Y E C •
We have actually something more: If X,Y E C , then [X,y] E C
This bracket product has the fallowing properties:
2
Y] - [Xl ' Y]
[Xl + X2 ' + [X2 ' Y ]
[X,y] .. - [Y,X]
[X, [y,z1] [Y, LZ ,XJ] [Z, (X,YJ J - 0,
+ +
E
for X, Y, Z C. Tho bracket is not bilinear over A, but only over
k. lIo havo
1
[X, ~
aY] (b) ... X (aY) - (aY) (X) (b)
• (Xa) (Yb) + a ( X, Y ] (b)
so that; [1..,aY] .. (x.1.) Y + c. (X,I] for X,YEC, a E A. The skew
conmutativity of tl-je t:-ackot gi"')S
[!LX,'l] .. -(Ya)X+a[X,Y].
When A i~ the a1pebra of differentiable functions on a
manifold, C is the s?9-c0 of differontiablo vector fields.
1.2 Derivation 190 ..; 3.
Definition 2. A deriva+-.1on law in a unitary A-module M is a map
D : C ~ Ho~ (M,M) such that, if DX denotes tho imago of X Ec
under this mar>, we h3.vo
1) DX + Y .. DX + Dy
Da,X - a DX for a E A, X,YEC •
i.e., D f: HomA (C, Ho~: (M,M».
3
U) Ox (au) .. (Xa) u + a Ox u for 0. E A, u EM.
In practice, M '<fill be the module of differentiable sections
of a vector bundle over a In[mifold V. A derivation law enllblos ono
thur. to dUforentiato sections of the bundlo in specified directions.
11' w(; consid0r A as'm A-modulo, then D clufined by
Ox a = Xa is a derivation law ill A. This will hereafter be referred
t.o as the canonical der'ivation in A • Moreover, if V is 3.ny modulo
over k, we may define on the A-module A ®V , a derivation law by
k
sotting 0X(a ® v) = Xa®v and extending by linearity. This shall also
be termed the canonical u"ri va ticn in A® V.
k
There exist modules which do not admit any derivation law.
For instance, let A be the algebra k [t] of polynomials in one vari
able t over k; then C is easily seen to be the free A-modulo
generated by F = 2> lot .. Let M be the •• -module AI tr. where tL
is tho ideal of polynomials without; constant term. If there wtlro3.
derivntion law in this moduie, dt'::notine by e the identity cosut of
AI tt , we have
o = Op (to) = (P.t)~ + t.Dp6 • ~,
which is a contradiction.
However, tho situation becomes better if we cOD!ino oversolves
to free A-modules.
l'hoorem 1.
Let M be a free A-modulo, (ei ) iEI being a. basis. Given