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 ${\displaystyle \iota _{X}\omega }$ is sometimes written as ${\displaystyle X\mathbin {\lrcorner } \omega .}$^[1]

Definition

The interior product is defined to be the contraction of a differential form with a vector field. Thus if ${\displaystyle X}$ is a vector field on the manifold ${\displaystyle M,}$ then

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

${\displaystyle p}$

${\displaystyle \omega }$

${\displaystyle (p-1)}$

${\displaystyle \iota _{X}\omega }$

$${\displaystyle (\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)}$$

${\displaystyle X_{1},\ldots ,X_{p-1}.}$

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

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

${\displaystyle \langle \,\cdot ,\cdot \,\rangle }$

${\displaystyle \alpha }$

${\displaystyle X.}$

${\displaystyle \beta }$

${\displaystyle p}$

${\displaystyle \gamma }$

${\displaystyle q}$

$${\displaystyle \iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).}$$

Properties

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

$${\displaystyle \iota _{X}(dx_{1}\wedge ...\wedge dx_{n})=\sum _{r=1}^{n}(-1)^{r-1}f_{r}dx_{1}\wedge ...\wedge {\widehat {dx_{r))}\wedge ...\wedge dx_{n},}$$

${\displaystyle dx_{1}\wedge ...\wedge {\widehat {dx_{r))}\wedge ...\wedge dx_{n))$

${\displaystyle dx_{r))$

${\displaystyle dx_{1}\wedge ...\wedge dx_{n))$

By antisymmetry of forms,

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

${\displaystyle \iota _{X}\circ \iota _{X}=0.}$

${\displaystyle d,}$

${\displaystyle d\circ d=0.}$

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^[2] or Cartan magic formula):

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

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.^[3] The Cartan homotopy formula is named after Élie Cartan.^[4]

The interior product with respect to the commutator of two vector fields ${\displaystyle X,}$ ${\displaystyle Y}$ satisfies the identity

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