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].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Hindustan Book Agency
About this chapter
Cite this chapter
Mukherjee, A. (2013). Characteristic Classes and Chern-Weil Construction. In: Atiyah-Singer Index Theorem. Texts and Readings in Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-60-6_5
Download citation
DOI: https://doi.org/10.1007/978-93-86279-60-6_5
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-93-80250-54-0
Online ISBN: 978-93-86279-60-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)