Insecurity for compact surfaces of positive genus

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Bangert, Victor
Gutkin, Eugene
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Abstract
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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.
Comment: 37 pages, 11 figures
Keywords
Mathematics - Dynamical Systems, Mathematics - Differential Geometry, 53C22, 57M10, 37E40
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