Integral of inverse functions

In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map from the Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$ to the space of differential forms on M that is equivariant; i.e., $\alpha (gX)=g\alpha (X)$. For an equivariant differential form $\alpha$, the equivariant exterior derivative $d_{\mathfrak{g}}\alpha$ of $\alpha$ is defined by

$(d_{\mathfrak{g}}\alpha )(X)=d(\alpha (X))-i_{X^{\#}}(\alpha (X))$

where d is the usual exterior derivative and $i_{X^{\#}}$ is the interior product by the fundamental vector field generated by X. It is easy to see $d_{\mathfrak{g}}\circ d_{\mathfrak{g}}=0$ (use the fact the Lie derivative of $\alpha (X)$ along $X^{\#}$ is zero) and this makes the space of equivariant differential forms a complex. One can then put

$H_G^{*}(X)=\ker d_{\mathfrak{g}}/\mathrm{im}{d}_{\mathfrak{g}}$,

which is called the equivariant cohomology of M. The definition is due to H. Cartan. The notion has an application to the equivariant index theory.

$d_{\mathfrak{g}}$-closed or $d_{\mathfrak{g}}$-exact forms are often called equivariantly closed or equivariantly exact.

References

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