  A pair of points in a riemannian manifold $M$ is secure if the geodesics
between the points can be blocked by a finite number of point obstacles;
otherwise the pair of points is insecure. A manifold is secure if all pairs of
points in $M$ are secure. A manifold is insecure if there exists an insecure
point pair, and totally insecure if all point pairs are insecure.
  Compact, flat manifolds are secure. A standing conjecture says that these are
the only secure, compact riemannian manifolds. We prove this for surfaces of
genus greater than zero. We also prove that a closed surface of genus greater
than one with any riemannian metric and a closed surface of genus one with
generic metric are totally insecure.
