Interior product

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]

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

The interior product with respect to the commutator of two vector fields ${\displaystyle X,}$ ${\displaystyle Y}$ satisfies the identity $${\displaystyle \iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right]=\left[\iota _{X},{\mathcal {L}}_{Y}\right].}$$Proof. For any k-form ${\displaystyle \Omega }$, $${\displaystyle {\mathcal {L}}_{X}(\iota _{Y}\Omega )-\iota _{Y}({\mathcal {L}}_{X}\Omega )=({\mathcal {L}}_{X}\Omega )(Y,-)+\Omega ({\mathcal {L}}_{X}Y,-)-({\mathcal {L}}_{X}\Omega )(Y,-)=\iota _{{\mathcal {L}}_{X}Y}\Omega =\iota _{[X,Y]}\Omega }$$and similarly for the other result.

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

where the anticommutator was used. 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]

• Cap product – Method in algebraic topology
• Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
• Tensor contraction – Operation in mathematics and physics

• Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011
• Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011. doi:10.1007/978-1-4419-7400-6