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[1] For an equivariant differential form
, the equivariant exterior derivative
is defined by where d is the usual exterior derivative and
is the interior product by the fundamental vector field generated by X.
(use the fact the Lie derivative of
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 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.
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