Interior product

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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; this character is U+2A3C in Unicode.


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


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 equipped with 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

See also


  • Theodore Frankel (2011) The Geometry of Physics: An Introduction Cambridge University Press