Characteristic classes for $G$-structures

Abstract

Let $G\subset GL(V)$ be a linear Lie group with Lie algebra $\mathfrak{g}$ and let $A(\mathfrak{g}{)}^G$ be the subalgebra of $G$-invariant elements of the associative supercommutative algebra $A(\frak g)= S(\frak g^*)\otimes \La(V^*)$. To any $G$-structure $\pi :P\to M$ with a connection $\omega$ we associate a homomorphism ${\mu }_{\omega }:A(\mathfrak{g}{)}^G\to \Omega (M)$. The differential forms ${\mu }_{\omega }(f)$ for $f\in A(\mathfrak{g}{)}^G$ which are associated to the $G$-structure $\pi$ can be used to construct Lagrangians. If $\omega$ has no torsion the differential forms ${\mu }_{\omega }(f)$ are closed and define characteristic classes of a $G$-structure. The induced homomorphism $μ'_ω:A(\g)^G\to H^*(M)$ does not depend on the choice of the torsionfree connection $\omega$ and it is the natural generalization of the Chern Weil homomorphism.