Sufficient conditions for periodicity of a Killing vector field

Abstract

Let X X be a complete Killing vector field on an n n -dimensional connected Riemannian manifold. Our main purpose is to show that if X X has as few as n n closed orbits which are located properly with respect to each other, then X 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 M . We give a generalization of this characterization which applies to arbitrary complete vector fields on M M .