Total Curvature of Curves in Riemannian Manifolds

Abstract

The total curvature of C2 curves embedded in an arbitrary Riemannian manifold is shown to be the limit of the curvatures of inscribed geodesic polygons. The formula for the total curvature of a curve as the least upper bound of curvatures of inscribed geodesic polygons holds for a manifold of non-positive sectional curvature only.