Interior product


In mathematics, the interior product is a degree −1 derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product ιXω is sometimes written as.

Definition

The interior product is defined to be the contraction of a differential form with a vector field. Thus if X is a vector field on the manifold M, then
is the map which sends a p-form ω to the -form ιXω defined by the property that
for any vector fields X1,..., Xp−1.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α
where is the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then
The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is called a derivation.

Properties

By antisymmetry of forms,
and so. This may be compared to the exterior derivative d, which has the property.
The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula :
This identity defines a duality between the exterior and interior derivatives. Cartan's identity is important in symplectic geometry and general relativity: see moment map. The Cartan homotopy formula is named after Élie Cartan.
The interior product with respect to the commutator of two vector fields, satisfies the identity