In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)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 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 \omega to the (p - 1)-form defined by the property that for any vector fields When \omega is a scalar field (0-form), by convention. The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms \alpha where is the duality pairing between \alpha and the vector X. Explicitly, if \beta is a p-form and \gamma 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

If in local coordinates the vector field X is given by then the interior product is given by where is the form obtained by omitting dx_r from. By antisymmetry of forms, and so This may be compared to the exterior derivative d, which has the property The interior product with respect to the commutator of two vector fields X, Y satisfies the identity Proof. For any k-form \Omega, and similarly for the other result.

Cartan identity

The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula or Cartan magic formula) : where the anticommutator was used. 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.