Insecurity for compact surfaces of positive genus
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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.
KeywordsCompact riemannian surface Minimal geodesic Closed geodesic Free homotopy class Blocking set Insecurity Flat torus Surface of genus greater than one
Mathematics Subject Classification (2000)53C22 57M10 37E40
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