Equivariant differential form

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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 to the space of differential forms on M that is equivariant; i.e.,

In other words, an equivariant differential form is an invariant element of [1]

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 (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.

-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.


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