THE FARY-MILNOR THEOREM IN HADAMARD MANIFOLDS

The Fary-Milnor theorem is generalized: Let gamma be a simple closed c urve in a;complete simply connected Riemannian S-manifold of nonpositi ve sectional curvature. If gamma has total curvature less than or equa l to 4 pi, then gamma is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a g eodesic segment shows that the bound 4 pi is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and th e proof by integral geometry did not apply to spaces of variable curva ture. Now, instead, a combinatorial proof has been devised.