The diameter of orbits of compact groups of isometries; Newman’s theorem for noncompact manifolds

Abstract:

The diameter of orbits of a compact isometry group G of a Riemannian manifold M cannot be uniformly small. If the sectional curvature of M is bounded above by ${b^2}$ (b real or pure imaginary), then explicit bounds are found for $D(M)$, where $D(M)$ is defined to be the largest number such that: If every orbit G has diameter less than $D(M)$, then G acts trivially on M. These bounds depend only on b and the injectivity radius of M. The proofs involve an investigation of various types of convex sets and an estimate for distance contraction of the exponential map on a manifold with bounded curvature.