Insecurity for compact surfaces of positive genus

TL;DR

This paper investigates the security of geodesic pairs on compact surfaces, proving that only flat surfaces of genus zero are secure, while higher genus surfaces are generally insecure or totally insecure.

Contribution

It proves the conjecture that only flat, genus-zero surfaces are secure, and demonstrates insecurity for higher genus surfaces with various metrics.

Findings

Compact flat manifolds are secure.

Surfaces of genus > 1 are totally insecure.

Genus 1 surfaces with generic metrics are totally insecure.

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.