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 $(k-1)$ form that maps $x\in M$ to $\omega_x(\xi_x, \cdot)$ In other words, $\omega$ is point-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. 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 $\eta$ \begin{eqnarray*} \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. \end{eqnarray*}
3. Contraction is an anti-derivation [1]. If $\omega^1$ is a $p$ form, and $\omega^2$ is a $q$ form, then $$ \iota_\xi \big(\omega^1\wedge \omega^2\big) = (\iota_\xi \omega^1) \wedge \omega^2 + (-1)^p\ \omega^1\wedge (\iota_\xi \omega^2).$$

Bibliography

1

T. Frankel, Geometry of physics, Cambridge University press, 1997.