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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Fary-Milnor theorem in Hadamard manifolds
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by Stephanie B. Alexander and Richard L. Bishop PDF
Proc. Amer. Math. Soc. 126 (1998), 3427-3436 Request permission

Abstract:

The Fary-Milnor theorem is generalized: Let $\gamma$ be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If $\gamma$ has total curvature less than or equal to $4\pi$, then $\gamma$ is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a geodesic segment shows that the bound $4\pi$ is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and the proof by integral geometry did not apply to spaces of variable curvature. Now, instead, a combinatorial proof has been devised.
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Additional Information
  • Stephanie B. Alexander
  • Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
  • Email: sba@math.uiuc.edu
  • Richard L. Bishop
  • Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
  • Email: bishop@math.uiuc.edu
  • Received by editor(s): October 2, 1996
  • Received by editor(s) in revised form: March 28, 1997
  • Communicated by: Christopher Croke
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3427-3436
  • MSC (1991): Primary 57M25; Secondary 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-98-04423-2
  • MathSciNet review: 1459103