Interior product

In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, 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 ${\displaystyle X\mathbin {\lrcorner } \omega .}$

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 $${\displaystyle \iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)}$$ is the map which sends a ${\displaystyle p}$-form ${\displaystyle \omega }$ to the ${\displaystyle (p-1)}$-form ${\displaystyle \iota _{X}\omega }$ defined by the property that $${\displaystyle (\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)}$$ for any vector fields ${\displaystyle X_{1},\ldots ,X_{p-1}.}$

When ${\displaystyle \omega }$ is a scalar field (0-form), ${\displaystyle \iota _{X}\omega =0}$ by convention.

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms ${\displaystyle \alpha }$ $${\displaystyle \displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle ,}$$ where ${\displaystyle \langle \,\cdot ,\cdot \,\rangle }$ is the duality pairing between ${\displaystyle \alpha }$ and the vector ${\displaystyle X.}$ Explicitly, if ${\displaystyle \beta }$ is a ${\displaystyle p}$-form and ${\displaystyle \gamma }$ is a ${\displaystyle q}$-form, then $${\displaystyle \iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).}$$ 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.

If in local coordinates ${\displaystyle (x_{1},...,x_{n})}$ the vector field ${\displaystyle X}$ is given by

${\displaystyle X=f_{1}{\frac {\partial }{\partial x_{1}}}+\cdots +f_{n}{\frac {\partial }{\partial x_{n}}}}$

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

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 ${\displaystyle d,}$ which has the property ${\displaystyle d\circ d=0.}$