Local Coordinate Description of Fibre Bundles

Abstract

In the first section, we show that, up to isomorphism, vector bundles are just locally trivial fibre bundles with a finite-dimensional vector space V as fibre and GL(V), the group of automorphisms of V, as a structure group. This is done by examining how trivial bundles are pieced together, using systems of transition functions to define a general locally trivial fibre bundle. We can apply this analysis to prove a theorem which says that any continuous functorial operation on vector spaces determines an operation on vector bundles. This allows construction of tensor products, exterior products, etc., of vector bundles.