# 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 $(k-1)$-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. 1.

For any real number $k$

 $\iota_{k\xi}\omega=k\iota_{\xi}\omega.$
2. 2.

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

 $\displaystyle\iota_{\xi+\eta}\omega$ $\displaystyle=$ $\displaystyle\iota_{\xi}\omega+\iota_{\eta}\omega,$ $\displaystyle\iota_{\xi}\iota_{\eta}\omega$ $\displaystyle=$ $\displaystyle-\iota_{\eta}\iota_{\xi}\omega,$ $\displaystyle\iota_{\xi}\iota_{\xi}\omega$ $\displaystyle=$ $\displaystyle 0.$
3. 3.

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

 $\iota_{\xi}\big{(}\omega^{1}\wedge\omega^{2}\big{)}=(\iota_{\xi}\omega^{1})% \wedge\omega^{2}+(-1)^{p}\ \omega^{1}\wedge(\iota_{\xi}\omega^{2}).$

## References

• 1 T. Frankel, Geometry of physics, Cambridge University press, 1997.
Title contraction Contraction 2013-03-22 13:37:28 2013-03-22 13:37:28 mathcam (2727) mathcam (2727) 4 mathcam (2727) Definition msc 15A75 msc 58A10