Besicovitch Covering Lemma, Hadamard manifolds, and zero entropy

It is proved that if the Besicovitch Covering Lemma is true on either a Hadamard manifold or a simply connected surface without focal points that covers a compact quotient, then the manifold is the Euclidean space. As a corollary, the vanishing of the topological entropy of a compact manifold of nonpositive curvature or of a compact surface without focal points is equivalent to the validity of the Besicovitch Covering Lemma on the universal covering space of the manifold.