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VOL. 34-2 | 1996 Riemannian geometry and mathematical physics: vector bundles and gauge theories
Michael K. Murray

Editor(s) Tim Cranny, John Hutchinson


The mathematical motivation for studying vector bundles comes from the example of the tangent bundle $TM$ of a manifold $M$. Recall that the tangent bundle is the union of all the tangent spaces $T_mM$ for every $m$ in $M$. As such it is a collection of vector spaces, one for every point of $M$.

The physical motivation comes from the realisation that the fields in physics may not just be maps $\phi : M \rightarrow C^N$ say, but may take values in different vector spaces at each point. Tensors do this for example. The argument for this is partly quantum mechanics because, if $\phi$ is a wave function on a space-time $M$ say, then what we can know about are expectation values, that is things like: \[\int_M \langle\phi(x), \phi(x)\rangle dx \] and to define these all we need to know is that $\phi(x)$ takes its values in a one-dimensional complex vector space with Hermitian inner product. There is no reason for this to be the same one-dimensional Hermitian vector space here as on Alpha Centuari. Functions like $\phi$, which are generalisations of complex valued functions, are called sections of vector bundles.

We will consider first the simplest theory of vector bundles where the vector space is a one-dimensional complex vector space -line bundles.


Published: 1 January 1996
First available in Project Euclid: 18 November 2014

zbMATH: 0849.53021
MathSciNet: MR1394690

Rights: Copyright © 1996, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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