# Interior product

In mathematics, the interior product (a.k.a. interior derivative, interior multiplication, inner multiplication, inner derivative, or inner derivation) 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, should not be confused with an inner product. The interior product ιXω is sometimes written as Xω.[1]

## 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

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

is the map which sends a p-form ω to the (p−1)-form ιXω defined by the property that

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

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

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

where ⟨ , ⟩ is the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then

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

The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is often called a derivative.

## Properties

By antisymmetry of forms,

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

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

The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (a.k.a. Cartan identity, Cartan homotopy formula[2] or Cartan magic formula):

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

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

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