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 [math]\displaystyle{ \iota_X \omega }[/math] is sometimes written as [math]\displaystyle{ X \mathbin{\lrcorner} \omega. }[/math]

Definition

The interior product is defined to be the contraction of a differential form with a vector field. Thus if [math]\displaystyle{ X }[/math] is a vector field on the manifold [math]\displaystyle{ M, }[/math] then [math]\displaystyle{ \iota_X : \Omega^p(M) \to \Omega^{p-1}(M) }[/math] is the map which sends a [math]\displaystyle{ p }[/math]-form [math]\displaystyle{ \omega }[/math] to the [math]\displaystyle{ (p - 1) }[/math]-form [math]\displaystyle{ \iota_X \omega }[/math] defined by the property that [math]\displaystyle{ (\iota_X\omega)\left(X_1, \ldots, X_{p-1}\right) = \omega\left(X, X_1, \ldots, X_{p-1}\right) }[/math] for any vector fields [math]\displaystyle{ X_1, \ldots, X_{p-1}. }[/math]

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

[math]\displaystyle{ X = f_1 \frac{\partial}{\partial x_1} + \cdots + f_r \frac{\partial}{\partial x_r} }[/math]

then the interior product is given by [math]\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, }[/math] where [math]\displaystyle{ dx_1\wedge ...\wedge \widehat{dx_r} \wedge ... \wedge dx_n }[/math] is the form obtained by omitting [math]\displaystyle{ dx_r }[/math] from [math]\displaystyle{ dx_1 \wedge ...\wedge dx_n }[/math].

By antisymmetry of forms, [math]\displaystyle{ \iota_X \iota_Y \omega = - \iota_Y \iota_X \omega, }[/math] and so [math]\displaystyle{ \iota_X \circ \iota_X = 0. }[/math] This may be compared to the exterior derivative [math]\displaystyle{ d, }[/math] which has the property [math]\displaystyle{ d \circ d = 0. }[/math]

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): [math]\displaystyle{ \mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\{ d, \iota_X \right\} \omega. }[/math]

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 [math]\displaystyle{ X, }[/math] [math]\displaystyle{ Y }[/math] satisfies the identity [math]\displaystyle{ \iota_{[X,Y]} = \left[\mathcal{L}_X, \iota_Y\right]. }[/math]

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