In mathematics, the interior product is a degree −1 antiderivation on the exterior algebra of differential forms on a smooth manifold. It 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 iXω defined by the property that
for any vector fields X1,..., Xp−1.
The interior product is also called interior or inner multiplication, or the inner derivative or derivation. The interior product ιX ω is sometimes written as X ⨼ ω; this chracter is U+2A3C in Unicode.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α
- ιXα = α(X),
the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then
Properties
By antisymmetry,
and so 
. The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan's identity:
This identity is important in symplectic geometry: see moment map.
See also
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