## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Instructional Workshop on Analysis and Geometry, Part 2. Tim Cranny and John Hutchinson, eds. Proceedings of the Centre for Mathematics and its Applications, v. 34. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1996), 165 - 184

### Riemannian geometry and mathematical physics: vector bundles and gauge theories

#### Abstract

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.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416323492

**Mathematical Reviews number (MathSciNet)**

MR1394690

**Zentralblatt MATH identifier**

0849.53021

#### Citation

Murray, Michael K. Riemannian geometry and mathematical physics: vector bundles and gauge theories. Instructional Workshop on Analysis and Geometry, Part 2, 165--184, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1996. https://projecteuclid.org/euclid.pcma/1416323492