Interior product

In mathematics, the interior product or interior derivative is a degree −1 antiderivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, is also called interior or inner multiplication, or the inner derivative or derivation, but should not be confused with an inner product. The interior product ι_Xω is sometimes written as X ⨼ ω.^[1]

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

$\iota_X\colon \Omega^p(M) \to \Omega^{p-1}(M)$

is the map which sends a p-form ω to the (p−1)-form ι_Xω defined by the property that

$( \iota_X\omega )(X_1,\ldots,X_{p-1})=\omega(X,X_1,\ldots,X_{p-1})$

for any vector fields X₁, ..., X_p−1.

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α

$\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$,

the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then

$\iota_X(\beta\wedge\gamma) = (\iota_X\beta)\wedge\gamma+(-1)^p\beta\wedge(\iota_X\gamma).$

The above relation says that the interior product obeys a graded Leibniz rule. An operation equipped with linearity and a Leibniz rule is often called a derivative.

Properties

By antisymmetry of forms,

$\iota_X \iota_Y \omega = - \iota_Y \iota_X \omega$

and so $\iota_X^2 = 0$. This may be compared to the exterior derivative d, which has the property d² = 0. The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan's identity:

$\mathcal L_X\omega = \mathrm d (\iota_X \omega) + \iota_X \mathrm d\omega.$

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 interior product with respect to the commutator of two vector fields $X$, $Y$ satisfies the identity

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

See also

• Cap product
• Inner product
• Tensor contraction

References

1. The character ⨼ is U+2A3C in Unicode

• Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011