You are not currently logged in.
Access your personal account or get JSTOR access through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
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
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/119165
Page Count: 10
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available
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.
Proceedings of the American Mathematical Society © 1998 American Mathematical Society