dg.differential geometry – Vector fields whose flows have constant norms


$newcommand{tr}{operatorname{tr}}$

Let $(M,g)$ be a smooth two-dimensional Riemannian manifold with boundary, and let $X$ be a vector field on $M$. Let $psi_t:M to M$ be the flow of $X$.

Suppose that $|(dpsi_t)_p|^2=langle (dpsi_t)_p,(dpsi_t)_p rangle=tr_g ((dpsi_t)_p^T(dpsi_t)_p)$ is independent of $p$, but not constant in $t$. Must $X$ be a homothetic vector field? i.e. does $L_x g=lambda g$ for some constant $lambda$? (In that case $psi_t^*g=e^{lambda t}g$. $lambda=0$ corresponds to Killing fields).

Since $f(t,p):=|(dpsi_t)_p|^2=tr_gbig((psi_t^*g)_pbig)$
a necessary and sufficient condition on $X$ is that $frac{partial }{partial t}f(t,p)=tr_g(psi_t^*L_Xg)$ would be independent of $p$, for all $t$.

In particular, for $t=0$, $tr_g(L_Xg)=2text{div}(X)$ should be constant.

So, a necessary (but insufficient) condition is for the divergence of $X$ to be constant.