Growth of the number of geodesics between points and insecurity for Riemannian manifolds

A Riemannian manifold is said to be uniformly secure if there is a finite number s such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by s point obstacles. We prove that the number of geodesics with length ≤ T between every pair of points in a uniformly secure manifold grows polynomially as T → ∞. By results of Gromov and Mañé, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat. This follows from the virtual nilpotency of its fundamental group either via the theorems of Croke-Schroeder and Burago-Ivanov, or by more recent work of Lebedeva. We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.