Besicovitch covering lemma, Hadamard manifolds, and zero entropy (Q1177617)

Suppose \(M\) is a Riemannian manifold covering a compact quotient, \(\dim M=n\). Suppose either that \(M\) is a Hadamard manifold (\(M\) has nonpositive curvature and is simply connected) or that \(\dim M=2\) and \(M\) is a simply connected surface without focal points. It is proved that under these conditions the assertions of the Besicovitch Covering Lemma (BCL) hold on \(M\) if and only if M is isometric to \(\mathbb{R}^ n\) (\(\mathbb{R}^ 2\), resp.). As a corollary the following is deduced: the topological entropy of the geodesic flow on a compact manifold with nonpositive curvature without focal points vanishes if and only if the assertions of the BCL hold on the universal cover of the manifold.