Interior product

Short description: Mapping from p forms to p-1 forms

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 ${\iota }_X\omega$ is sometimes written as $X⌟\omega .$

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 $${\iota }_X:{\mathrm{\Omega }}^p(M)\to {\mathrm{\Omega }}^{p-1}(M)$$ is the map which sends a $p$-form $\omega$ to the $(p-1)$-form ${\iota }_X\omega$ defined by the property that $$({\iota }_X\omega )(X_1,\dots ,X_{p-1})=\omega (X,X_1,\dots ,X_{p-1})$$ for any vector fields $X_1,\dots ,X_{p-1}.$

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms $\alpha$ $${\iota }_X\alpha =\alpha (X)=⟨\alpha ,X⟩,$$ where $⟨\cdot ,\cdot ⟩$ is the duality pairing between $\alpha$ and the vector $X.$ Explicitly, if $\beta$ is a $p$-form and $\gamma$ is a $q$-form, then $${\iota }_X(\beta \wedge \gamma )=\left({\iota }_X\beta \right)\wedge \gamma +(-1{)}^p\beta \wedge \left({\iota }_X\gamma \right).$$ 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 $(x_1,...,x_n)$ the vector field $X$ is given by

$X=f_1\frac{\partial }{\partial x_1}+\cdots +f_r\frac{\partial }{\partial x_r}$

then the interior product is given by $${\iota }_X(d{x}_1\wedge ...\wedge d{x}_n)=\sum _{r=1}^n(-1{)}^{r-1}{f}_{r}d{x}_1\wedge ...\wedge \widehat{d{x}_r}\wedge ...\wedge d{x}_n,$$ where $d{x}_1\wedge ...\wedge \widehat{d{x}_r}\wedge ...\wedge d{x}_n$ is the form obtained by omitting $d{x}_r$ from $d{x}_1\wedge ...\wedge d{x}_n$.

By antisymmetry of forms, $${\iota }_X{\iota }_Y\omega =-{\iota }_Y{\iota }_X\omega ,$$ and so ${\iota }_X\circ {\iota }_X=0.$ This may be compared to the exterior derivative $d,$ which has the property $d\circ d=0.$

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 or Cartan magic formula): $${{ℒ}}_X\omega =d({\iota }_X\omega )+{\iota }_{X}d\omega =\{d,{\iota }_X\}\omega .$$

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. The Cartan homotopy formula is named after Élie Cartan.

The interior product with respect to the commutator of two vector fields $X,$ $Y$ satisfies the identity $${\iota }_{[X,Y]}=[{{ℒ}}_X,{\iota }_Y].$$

See also

• Cap product
• Tensor contraction – Operation in mathematics and physics

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