Sufficient Conditions for Periodicity of a Killing Vector Field

Abstract

Let $X$ be a complete Killing vector field on an $n$-dimensional connected Riemannian manifold. Our main purpose is to show that if $X$ has as few as $n$ closed orbits which are located properly with respect to each other, then $X$ must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold $M$. We give a generalization of this characterization which applies to arbitrary complete vector fields on $M$.