The security conjecture for Riemannian tori

Let $(M,g)$ be a Riemannian torus, meaning a manifold diffeomorphic to a torus equipped with a Riemannian metric. A finite set $B\subset M\setminus\{x,y\}$ is a blocking set for points $x,y\in M$ if every geodesic joining $x$ and $y$ passes through a point of $B$. The torus $(M,g)$ is secure if every pair of points, including a point paired with itself, admits such a finite blocking set. Torus security conjecture. A Riemannian torus is secure if and only if it is flat. The abstract describes this as a consequence of the general security conjecture, and the paper proves it for two-dimensional tori; the assertion for tori of arbitrary dimension remains open in the source.