# Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map

$\alpha :{\mathfrak {g}}\to \Omega ^{*}(M)$ from the Lie algebra ${\mathfrak {g}}=\operatorname {Lie} (G)$ to the space of differential forms on M that is equivariant; i.e.,

$\alpha (\operatorname {Ad} (g)X)=g\alpha (X).$ In other words, an equivariant differential form is an invariant element of $\mathbb {C} [{\mathfrak {g}}]\otimes \Omega ^{*}(M).$ $(d_{\mathfrak {g}}\alpha )(X)=d(\alpha (X))-i_{X^{\#}}(\alpha (X))$ $H_{G}^{*}(X)=\operatorname {ker} d_{\mathfrak {g}}/\operatorname {im} d_{\mathfrak {g}}$ ,

which is called the equivariant cohomology of M (which coincides with the ordinary equivariant cohomology defined in terms of Borel construction.) The definition is due to H. Cartan. The notion has an application to the equivariant index theory.

The integral of an equivariantly closed form may be evaluated from its restriction to the fixed point by means of the localization formula.