About every convex set in any generic Riemannian manifold

Abstract

Abstract We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold ( M , g ) {(M,g)} . For example, if a convex set in ( M , g ) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex.