Interior product

From Wikipedia, the free encyclopedia

In mathematics, the interior product is a degree −1 derivation on the exterior algebra of differential forms on a smooth manifold. It 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 iXω 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,..., Xp−1.

The interior product is also called interior or inner multiplication, or the inner derivative or derivation.

The interior product is sometimes written as X \llcorner \omega = \iota_X\omega .

[edit] Properties

By antisymmetry,

ιXιYω = − ιYιXω

and so \iota_X^2 = 0 .

The interior product is related to the exterior product and the 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 is important in symplectic geometry: see moment map.

[edit] See also

 This geometry-related article is a stub. You can help Wikipedia by expanding it.
Languages