The Fary-Milnor theorem in Hadamard manifolds

Abstract

The Fary-Milnor theorem is generalized: Letγ\gammabe a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. Ifγ\gammahas total curvature less than or equal to4π4\pi, thenγ\gammais 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 bound4π4\piis 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.