# Equivariant differential form

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.

## References

- ↑ Proof: with , we have:

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