Interior product

In mathematics, the interior product (aka interior derivative, interior multiplication, inner multiplication, inner derivative, or inner derivation) is a degree 1 antiderivation 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 Xω.[1]

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 (p1)-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 often called a derivative.

Properties

By antisymmetry of forms,

and so . This may be compared to the exterior derivative d, which has the property d2 = 0.

The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan formula (a.k.a. Cartan identity, Cartan homotopy formula[2] or Cartan magic 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. (There is another formula called "Cartan formula". See Steenrod algebra.) Cartan homotopy formula is named after Élie Cartan.[3]

The interior product with respect to the commutator of two vector fields , satisfies the identity