The Fary-Milnor theorem in Hadamard manifolds

Abstract

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