# Interior product

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 *ι*_{X}*ω* is sometimes written as *X* ⨼ *ω*; this character is U+2A3C in Unicode.

## Definition

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 *ι*_{X}*ω* defined by the property that

for any vector fields *X*_{1},..., *X*_{p−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.

## Properties

By antisymmetry of forms,

and so . This may be compared to the exterior derivative *d* which has the property *d*^{2} = 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

## References

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