Topological properties of manifolds admitting a Yx-Riemannian metric

A complete Riemannian manifold (M,g) is a Ylx-manifold if every unit speed geodesic γ(t) originating at γ(0)=x∈M satisfies γ(l)=x for 0≠l∈R. Bérard-Bergery proved that if (M^m,g),m>1 is a Ylx-manifold, then M is a closed manifold with finite fundamental group, and the cohomology ring H^∗(M,Q) is generated by one element. We say that (M,g) is a Y^x-manifold if for every ∊>0 there exists l>∊ such that for every unit speed geodesic γ(t) originating at x, the point γ(l) is ∊-close to x. We use Low's notion of refocussing Lorentzian space-times to show that if (M^m,g),m>1 is a Y^x-manifold, then M is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a Y^x-manifold is a Y^x-manifold. Another corollary is that if (M^m,g),m=2,3 is a Y^x-manifold, then (M,h) is a Ylx-manifold for some metric h.