interior product

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 $\vec{v}\,$ is a vector field on the manifold $M\,$ , then

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

is the map which sends a p-form $\omega\,$ to the $p-1\,$ -form $i_v\omega\,$ defined by the property that

$(i_v\omega)(u_1,\ldots,u_{p-1})=\omega(v,u_1,\ldots,u_{p-1})$

for any vector fields $u_1...u_{p-1}\,$ .

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

Properties

$i_u i_v\omega = -i_v i_u\omega\,$ .

$i_v i_v\omega = 0\,$ .