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 &minus;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 $$X \mathbin{\lrcorner} \omega.$$

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

The interior product is the unique antiderivation of degree &minus;1 on the exterior algebra such that on one-forms $$\alpha$$ $$\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha, X \rangle,$$ where $$\langle \,\cdot, \cdot\, \rangle$$ 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

$$X = f_1 \frac{\partial}{\partial x_1} + \cdots + f_n \frac{\partial}{\partial x_n} $$

then the interior product is given by $$\iota_X (dx_1 \wedge ...\wedge dx_n) = \sum_{r=1}^{n}(-1)^{r-1}f_r dx_1 \wedge ...\wedge \widehat{dx_r} \wedge ... \wedge dx_n,$$ where $$dx_1\wedge ...\wedge \widehat{dx_r} \wedge ... \wedge dx_n$$ is the form obtained by omitting $$dx_r$$ from $$dx_1 \wedge ...\wedge dx_n$$.

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) : $$\mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\{ d, \iota_X \right\} \omega.$$

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. 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 $$\iota_{[X,Y]} = \left[\mathcal{L}_X, \iota_Y\right].$$