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## Summary

In mathematics, the interior product (also known as 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 $\iota _{X}\omega$ is sometimes written as $X\mathbin {\lrcorner } \omega .$ ## 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

$\iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)$

is the map which sends a $p$ -form $\omega$  to the $(p-1)$ -form $\iota _{X}\omega$  defined by the property that
$(\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)$

for any vector fields $X_{1},\ldots ,X_{p-1}.$

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms $\alpha$

$\displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle ,$

where $\langle \,\cdot ,\cdot \,\rangle$  is the duality pairing between $\alpha$  and the vector $X.$  Explicitly, if $\beta$  is a $p$ -form and $\gamma$  is a $q$ -form, then
$\iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).$

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.

## Properties

By antisymmetry of forms,

$\iota _{X}\iota _{Y}\omega =-\iota _{Y}\iota _{X}\omega$

and so $\iota _{X}\circ \iota _{X}=0.$  This may be compared to the exterior derivative $d,$  which has the property $d\circ d=0.$

The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula or Cartan magic formula):

${\mathcal {L}}_{X}\omega =d(\iota _{X}\omega )+\iota _{X}d\omega =\left\{d,\iota _{X}\right\}\omega .$

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 $X,$  $Y$  satisfies the identity

$\iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right].$