Insecurity for compact surfaces of positive genus

Details Insecurity for compact surfaces of positive genus Insecurity for compact surfaces of positive genus

2009-08-08

arxiv.org/abs/0908.1138v1

Geometriae Dedicata 146 (2010), 165 -- 191

Mathematics

Dynamical Systems

37 pages, 11 figures

Scientific paper

10.1007/s10711-009-9432-8

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.

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