Real Analytic Metrics on S^2 with Total Absence of Finite Blocking
  If (M,g) is a Riemannian manifold and x,y are points in M, then a subset P of M\{x,y} is said to be a blocking set for (x,y) if every geodesic from x to y passes through a point of P. If no pair (x,y) in M X M has a finite blocking set, then (M,g) is said to be totally insecure. We prove that there exist real analytic metrics h on S^2 such that (S^2,h) is totally insecure. 
Marlies Gerber and Lihuei Liu
math.DG