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The Fary-Milnor Theorem in Hadamard Manifolds

Stephanie B. Alexander and Richard L. Bishop
Proceedings of the American Mathematical Society
Vol. 126, No. 11 (Nov., 1998), pp. 3427-3436
Stable URL: http://www.jstor.org/stable/119165
Page Count: 10
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The Fary-Milnor Theorem in Hadamard Manifolds
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Abstract

The Fary-Milnor theorem is generalized: Let γ be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If γ has total curvature less than or equal to 4π , then γ 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 4π 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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