Equivariant differential form
In other words, an equivariant differential form is an invariant element of 
where d is the usual exterior derivative and is the interior product by the fundamental vector field generated by X. It is easy to see (use the fact the Lie derivative of along is zero) and one then puts
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.