The Chern–Weil homomorphism is a fundamental concept in differential geometry and algebraic topology that establishes a connection between characteristic classes of vector bundles and differential forms on manifolds. It provides a way to compute characteristic classes, which are topological invariants that classify vector bundles over a manifold, by using the curvature of connections on those bundles.
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The Chern-Weil homomorphism turns a polynomial in curvature into a topological cohomology class. For a principal bundle with Lie group , a degree- polynomial on its Lie algebra that is invariant under the adjoint action determineswhere is the curvature form of any principal connection. The Bianchi identity makes a closed differential form. Changing the principal connection changes it by an exact differential form, so the de Rham cohomology class depends on the principal bundle, not that choice. Products of polynomials become cup products.[ref]
Characteristic forms and geometric invariants by Shiing-Shen Chern and James Simons. Section 2, Theorem 2.12, page 51
For example, the trace-square polynomial produces , underlying the first Pontryagin class. This is the bridge between the locally measured curvature and a global topological invariant.