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 contraction of $\omega $ with $\xi $ is the smooth $(k1)$form that maps $x\in M$ to ${\omega}_{x}({\xi}_{x},\cdot )$. In other words, $\omega $ is pointwise evaluated with $\xi $ in the first slot. We shall denote this $(k1)$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. Then the 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 antiderivation [1]. If ${\omega}^{1}$ is a $p$form, and ${\omega}^{2}$ is a $q$form, then
$${\iota}_{\xi}\left({\omega}^{1}\wedge {\omega}^{2}\right)=({\iota}_{\xi}{\omega}^{1})\wedge {\omega}^{2}+{(1)}^{p}{\omega}^{1}\wedge ({\iota}_{\xi}{\omega}^{2}).$$
References
 1 T. Frankel, Geometry of physics, Cambridge University press, 1997.
Title  contraction 

Canonical name  Contraction 
Date of creation  20130322 13:37:28 
Last modified on  20130322 13:37:28 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  4 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 15A75 
Classification  msc 58A10 