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Let (A/, g) be a compact, connected, Riemannian manifold. Let X be a Killing vectot field on Ai. / = 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 / which enables us to prove facts relating D, the zero ponts of X, orbits of X which are closed geodesies, and, applying theorems of Klingenberg, the curvatute of M. Then we use these results together with a further analysis of / to describe the nature of a Killing vector field in a neighborhood of an isolated zero point.

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