Abstract
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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Bangert, V., Gutkin, E. Insecurity for compact surfaces of positive genus. Geom Dedicata 146, 165–191 (2010). https://doi.org/10.1007/s10711-009-9432-8
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DOI: https://doi.org/10.1007/s10711-009-9432-8
Keywords
- Compact riemannian surface
- Minimal geodesic
- Closed geodesic
- Free homotopy class
- Blocking set
- Insecurity
- Flat torus
- Surface of genus greater than one