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Proceedings of the American Mathematical Society

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

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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