The security–uniform security–flatness conjecture for closed Riemannian manifolds

Let $(M,g)$ be a closed $C^{\infty}$ Riemannian manifold. Here, $(M,g)$ is secure if every pair of points has the finite blocking property, and uniformly secure if there is a uniform bound on the cardinalities of blocking sets for all pairs of points. A metric $g$ is flat when its Riemann curvature tensor vanishes. The security–uniform security–flatness conjecture. The following statements are equivalent:

1. $(M,g)$ is secure.
2. $(M,g)$ is uniformly secure.
3. $g$ is a flat metric.

The conjecture relates the finite blocking property for geodesics to flatness. The cited discussion notes that results for surfaces support the implication from security to flatness, while the implication from flatness to the other conditions is established in earlier work; the full equivalence is presented as a conjecture.