Characteristic Classes and Connections

Abstract

Apart from the previous chapter, the theory of fibre bundles in this book is a theory over an arbitrary space. Even the relation to manifolds in Chapter 18 is treated from a topological point of view, but in the context of smooth manifolds and vector bundles we can approach Chern classes using constructions from analysis. This idea, which goes back to a letter from A. Weil (see A. Weil Collected papers, Volume III, pages 422–36 and 571–574), involves choosing a connection or covariant derivative on the complex vector bundle, defining the curvature 2-form of the connection, and representing the characteristic class as closed 2q-form which is a polynomial in the curvature form. This proceedure is outlined in this chapter.