Let $(M^n,g)$ be a Riemannian manifold that admit a unit Killing vector field $X$. i.e., $\mathscr{L}\_Xg=0$. Is it possible that there exist a smooth function $f$ on $M$ such that $X=\mathrm{grad}f$?


A good reference will be very appreciated.



When a Killing vector field on Riemannian manifold $(M,g)$ is gradient?
It is possible for this to happen (for example, the unit translation vector fields in $\mathbb{R}^n$), but it does not necessarily hold (for example, there are unit Killing vector fields on the stanard Riemannian $3$-sphere, but, obviously, none of them could be gradients, since they never vanish and the $3$-sphere is compact).


I'm not sure what fact or theory you want a reference for.


