Zero points of Killing vector fields, geodesic orbits, curvature, and cut locus

Abstract

Let $(M,g)$ be a compact, connected, Riemannian manifold. Let $X$ be a Killing vector field on $M$. $f = g(X,X)$ is called the length function of $X$. Let $D$ denote the minimum of the distances from points to their cut loci on $M$. We derive an inequality involving $f$ which enables us to prove facts relating $D$, the zero ponts of $X$, orbits of $X$ which are closed geodesics, and, applying theorems of Klingenberg, the curvature of $M$. Then we use these results together with a further analysis of $f$ to describe the nature of a Killing vector field in a neighborhood of an isolated zero point.