Characteristic Classes and Chern-Weil Construction

Abstract

In this chapter, we discuss the theory of characteristic classes. In the first part we present the axioms for Chern classes of a complex vector bundle, and then prove their existence and uniqueness. In the second part we introduce the notion of connection on a smooth vector bundle and curvature of connection, with related geometric concepts, including Riemannian or Levi-Civita connection and unitary connection, We then use Chern-Weil theory to construct Chern classes for a smooth complex vector bundle with a connection, as de Rham cohomology classes of the base space of the bundle, represented by invariant polynomials in the curvature of the connection. In the final section, we discuss Pontrjagin classes of a real vector bundle. The facts about fibre bundles which we use in the first part may be found in Chapter 1. Further details about these may be found in Steenrod [60]] or Husemoller [34].