Chern–Weil theory

In mathematics, the Chern -Weil theory is a general method how to calculate the characteristic classes of a principal bundle by its curvature. ( Characteristic classes are Kohomologieklassen that measure topologically how twisted a bundle. ) Historically, it originated in the proof of the higher dimensional version of the theorem of Gauss -Bonnet, it marked the beginning of the " global differential geometry ", ie the interaction of geometry and topology. The theory is named after André Weil and Chern SS.

Definition

Be a principal bundle with structure group the Lie algebra is of. Chern -Weil theory defines a homomorphism

The space of invariant polynomials on the deRham cohomology, the so-called Chern -Weil homomorphism.

Each invariant polynomial is the form

Associated with the curvature form of a connection of the principal bundle. That is, for

Is a closed form and is then by definition the cohomology class of this form. It can be shown that it is not dependent on the selected connection.

Examples

• Be. Then the curvature form has values ​​in. The development

Universal Chern -Weil homomorphism

Let be a Lie group and its classifying space. is not a manifold, nevertheless can be defined for the universal bundle of a Chern -Weil homomorphism.

If a principal bundle and its classifying map is, then.