Chern–Weil homomorphism

In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.

Let G be a real or complex Lie group with Lie algebra ${\displaystyle {\mathfrak {g}}}$; and let ${\displaystyle \mathbb {C} [{\mathfrak {g}}]}$ denote the algebra of ${\displaystyle \mathbb {C} }$-valued polynomials on ${\displaystyle {\mathfrak {g}}}$ (exactly the same argument works if we used ${\displaystyle \mathbb {R} }$ instead of ${\displaystyle \mathbb {C} }$.) Let ${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}}$ be the subalgebra of fixed points in ${\displaystyle \mathbb {C} [{\mathfrak {g}}]}$ under the adjoint action of G; that is, it consists of all polynomials f such that for any g in G and x in ${\displaystyle {\mathfrak {g}}}$, ${\displaystyle f(\operatorname {Ad} _{g}x)=f(x).}$

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