# 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.