# Divergence of left invariant vector field

by paweld
Tags: divergence, field, invariant, vector
 Sci Advisor HW Helper PF Gold P: 2,602 Given a volume form $\omega$, then the divergence of a vector field $X$ is related to the Lie derivative as: $(\mathrm{div}~X) \omega = \mathcal{L}_X \omega.$ In the case of a Lie group, there are presumably a number of ways to show that the RHS vanishes iff $X$ is an element of the Lie algebra. In particular, we can argue that it's natural from the point of view of the Lie algebra generating the isometries of the Lie group manifold.