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

If in local coordinates $(x_{1},...,x_{n})$ the vector field $X$ is described by functions $f_{1},...,f_{n)$ , then the interior product is given by

$\iota _{X}(dx_{1}\wedge ...\wedge dx_{n})=\sum _{r=1}^{n}(-1)^{r-1}f_{r}dx_{1}\wedge ...\wedge {\widehat {dx_{r))}\wedge ...\wedge dx_{n},$ where $dx_{1}\wedge ...\wedge {\widehat {dx_{r))}\wedge ...\wedge dx_{n)$ is the form obtained by omitting $dx_{r)$ from $dx_{1}\wedge ...\wedge dx_{n)$ .

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

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