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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: David Hoffman
Journal: Trans. Amer. Math. Soc. 233 (1977), 223-233
MSC: Primary 57E15
MathSciNet review: 0494171
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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.

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Keywords: Transformation group, convexity, orbit, Riemannian sectional curvature, Rauch comparison theorem
Article copyright: © Copyright 1977 American Mathematical Society