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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The diameter of orbits of compact groups of isometries; Newman’s theorem for noncompact manifolds
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by David Hoffman PDF
Trans. Amer. Math. Soc. 233 (1977), 223-233 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 233 (1977), 223-233
  • MSC: Primary 57E15
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0494171-0
  • MathSciNet review: 0494171