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 real or pure imaginary), then explicit bounds are found for , where is defined to be the largest number such that: If every orbit G has diameter less than , 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.
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Keywords: Transformation group, convexity, orbit, Riemannian sectional curvature, Rauch comparison theorem
Article copyright: © Copyright 1977 American Mathematical Society