2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167696The blocking number of a manifold is the minimal number of points needed to block out lights between any two given points in the manifold. It has been conjectured that if the blocking number of a manifold is finite, then the manifold must be flat. In this paper we prove that this is true for 2-dimensional manifolds with non-trivial fundamental groups.This is a very preliminary version of a paper about blocking numbers of compact Riemannian surfaces, the aim is to show that if the blocking number is finite, then the surface has to be flat. edit: similar results for 2-dimensional torus have been obtained by V. Bangert and E. Gutkin, reference to their paper has been added v3: minor changes with the referencesDifferential Geometry52C22text