contraction

Definition Let $\omega$ be a smooth $k$-form on a smooth manifold $M$,and let $\xi$ be a smooth vector field on $M$. The contractionof $\omega$ with $\xi$ is the smooth $(k-1)$-form that maps $x\in M$ to${\omega }_x({\xi }_x,\cdot )$.In other words, $\omega$ ispoint-wise evaluated with $\xi$ in the first slot.We shall denote this $(k-1)$-form by ${\iota }_{\xi }\omega$.If $\omega$ is a $0$-form, we set ${\iota }_{\xi }\omega =0$ for all $\xi$.

Properties Let $\omega$ and $\xi$ be as above. Thenthe following properties hold:

1. For any real number $k$

   $${\iota }_{k\xi }\omega =k{\iota }_{\xi }\omega .$$

2. For vector fields $\xi$ and $\eta$

   ${\iota }_{\xi +\eta }\omega$$=$${\iota }_{\xi }\omega +{\iota }_{\eta }\omega ,$

   ${\iota }_{\xi }{\iota }_{\eta }\omega$$=$$-{\iota }_{\eta }{\iota }_{\xi }\omega ,$

   ${\iota }_{\xi }{\iota }_{\xi }\omega$$=$$0.$

3. Contraction is an anti-derivation [1]. If${\omega }^1$ is a $p$-form, and ${\omega }^2$ is a $q$-form, then

   $${\iota }_{\xi }\left({\omega }^1\wedge {\omega }^2\right)=({\iota }_{\xi }{\omega }^1)\wedge {\omega }^2+{(-1)}^p{\omega }^1\wedge ({\iota }_{\xi }{\omega }^2).$$