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 X ⨼ ω.
is the map which sends a p-form ω to the (p−1)-form ιXω defined by the property that
for any vector fields X1, ..., Xp−1.
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.
By antisymmetry of forms,
and so . This may be compared to the exterior derivative d, which has the property d2 = 0.
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.
The interior product with respect to the commutator of two vector fields , satisfies the identity