Real Analytic Metrics on S^2 with Total Absence of Finite Blocking

TL;DR

This paper demonstrates the existence of real analytic metrics on the 2-sphere that make it totally insecure, meaning no finite set blocks all geodesics between any two points, challenging previous assumptions about geodesic security.

Contribution

It constructs explicit examples of real analytic metrics on S^2 that are totally insecure, showing such metrics are possible beyond smooth or generic cases.

Findings

Existence of real analytic totally insecure metrics on S^2

Construction method for such metrics

Implication for geodesic security theory

Abstract

If (M,g) is a Riemannian manifold and x,y are points in M, then a subset P of M\{x,y} is said to be a blocking set for (x,y) if every geodesic from x to y passes through a point of P. If no pair (x,y) in M X M has a finite blocking set, then (M,g) is said to be totally insecure. We prove that there exist real analytic metrics h on S^2 such that (S^2,h) is totally insecure.