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.
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 (p−1)-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.
By antisymmetry of forms,
and so . This may be compared to the exterior derivative d, which has the property d ∘ d = 0.
The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (a.k.a. Cartan identity, Cartan homotopy formula 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. The Cartan homotopy formula is named after Élie Cartan.
The interior product with respect to the commutator of two vector fields , satisfies the identity
- Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011
- Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011. doi:10.1007/978-1-4419-7400-6