In mathematics, the interior product or interior derivative 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, is also called interior or inner multiplication, or the inner derivative or derivation, but 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. The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan's identity:
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 interior product with respect to the commutator of two vector fields , satisfies the identity
- The character ⨼ is U+2A3C in Unicode
- Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011