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 be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If has total curvature less than or equal to , then 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 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

DOI:
https://doi.org/10.1090/S0002-9939-98-04423-2

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