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The Fary-Milnor theorem in Hadamard manifolds


Authors: Stephanie B. Alexander and Richard L. Bishop
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
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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

Keywords: Knots, total curvature, CAT(0) spaces, Hadamard manifolds
Received by editor(s): October 2, 1996
Received by editor(s) in revised form: March 28, 1997
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society