Suppose that $\Phi_t$ is a the global flow associated with a vector field $X$ on a Riemannian manifold $M$ and that $Y$ is any other vector field. Suppose furthermore that $X$ is a Killing vector field. Is there any way to write $$ \operatorname{div} [(\Phi_t)_* Y] $$ that is simpler than just writing it out in coordinates?
Thank you.
EDIT: What about $$ \nabla_{\Phi_*Y}(\Phi_* Z)? $$ (where $Z$ is a vector field)
Writing everything in coordinates should not be too bad. But I'll try without, writing $f$ instead of $\Phi_t$. Let $\mu$ be the volume form. Recall the coordinate-free formula for divergence $d(i_Y\mu)=(\operatorname{div} Y )\mu$. For any diffeomorphism $f$ we have $i_{f_*Y}((f^{-1})^*\mu) = (f^{-1})^* (i_Y\mu)$. Since $f$ is an isometry, $(f^{-1})^*\mu=\mu$. Thus, $i_{f_*Y}\mu = (f^{-1})^* (i_Y\mu)$. Applying $d$ to both sides, we get $$ (\operatorname{div} f_*Y) \mu = d(i_{f_*Y}\mu ) = d((f^{-1})^* (i_Y\mu)) = (f^{-1})^* (d (i_Y\mu)) = (f^{-1})^*((\operatorname{div} Y )\mu) $$ hence $\operatorname{div} f_*Y =(\operatorname{div} Y ) \circ f^{-1}$.