American Institute of Mathematical Sciences

June  2008, 21(2): 403-413. doi: 10.3934/dcds.2008.21.403

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

 1 Department of Mathematics, Northwestern University, Evanston, IL 60208-2730 2 IMPA, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil

Received  May 2007 Revised  October 2007 Published  March 2008

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 $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. 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.
Citation: Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403
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