Connections and Curvature

Abstract

There are many more classical applications of homotopy theory to geometry, but the central theme here being differential geometry, we wish to return to the differentiable category. Our next endeavor is to try and understand how bundles fail to be products by parallel translating vectors around closed loops. This depends of course on what is meant by “parallel translation” (which is explained in the section below), but roughly speaking, if parallel translation always results in the original vector, then the bundle is said to be flat. Otherwise, it is curved, and the amount of curvature is measured by the holonomy group, which tallies the difference between the end product in parallel translation and the original one. In this chapter, all maps and bundles are once again assumed to be differentiable.