In mathematics, the interior product (a.k.a. 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 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 called a derivation.
By antisymmetry of forms,
and so . This may be compared to the exterior derivative d, which has the property d ∘ d = 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. The Cartan homotopy formula is named after Élie Cartan.
The interior product with respect to the commutator of two vector fields , satisfies the identity