Minimal geodesics and nilpotent fundamental groups (Q1373236)

A non-constant geodesic \(c:\mathbb{R} \rightarrow M\) on a Riemannian manifold is called minimal if for all \(s \leq t\) the length of \(c|_{[s,t]}\) is not larger than the length of any curve homotopic to \(c|_{[s,t)}\) with fixed endpoints. \textit{V. Bangert} [Ergodic Theory Dyn. Syst. 10, 263-286 (1990; Zbl 0676.53055)] proved that the number of directions of minimal geodesics on a Riemannian manifold is at least the first Betti number \(b_1\). The author constructs compact Riemannian manifolds with only \(b_1\) different directions of minimal geodesics for arbitrary nilpotent fundamental groups. These examples show that Bangert's result is optimal in this case.