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 (use the fact the Lie derivative of along is zero) and one then puts