Interior product

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 :$iota\_Xcolon\; Omega^p(M)\; o\; Omega^\{p-1\}(M)$is the map which sends a "p"-form "ω" to the ("p"−1)-form "i"_"X""ω" defined by the property that:$(\; iota\_Xomega\; )(X\_1,ldots,X\_\{p-1\})=omega(X,X\_1,ldots,X\_\{p-1\})$for any vector fields "X"₁,..., "X"_"p"−1.

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

The interior product is sometimes written as $X\; llcorner\; omega\; =\; iota\_Xomega$.

Properties

By antisymmetry,

:$iota\_X\; iota\_Y\; omega\; =\; -\; iota\_Y\; iota\_X^\{\; \}\; omega$

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"::$mathcal\; L\_Xomega\; =\; mathrm\; d\; (iota\_X\; omega)\; +\; iota\_X\; mathrm\; domega.$This identity is important in symplectic geometry: see moment map.

ee also

* Tensor contraction