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

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

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

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.

Properties

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

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

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 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]

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

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

See also

• Cap product – method of adjoining a chain of with a cochainPages displaying wikidata descriptions as a fallback
• 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

Notes

References

• 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