Equivariant differential form


In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map
from the Lie algebra to the space of differential forms on M that are equivariant; i.e.,
In other words, an equivariant differential form is an invariant element of
For an equivariant differential form, the equivariant exterior derivative of is defined by
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 and one then puts
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.
-closed or -exact forms are called equivariantly closed or equivariantly exact.
The integral of an equivariantly closed form may be evaluated from its restriction to the fixed point by means of the localization formula.