Topological properties of manifolds admitting a YxYx-Riemannian metric

A complete Riemannian manifold (M,g)(M,g) is a Ylx-manifold if every unit speed geodesic γ(t)γ(t) originating at γ(0)=x∈Mγ(0)=x∈M satisfies γ(l)=xγ(l)=x for 0≠l∈R0≠l∈R. Bérard-Bergery proved that if (Mm,g),m>1(Mm,g),m>1 is a Ylx-manifold, then MM is a closed manifold with finite fundamental group, and the cohomology ring H∗(M,Q)H∗(M,Q) is generated by one element.We say that (M,g)(M,g) is a YxYx-manifold if for every ϵ>0ϵ>0 there exists l>ϵl>ϵ such that for every unit speed geodesic γ(t)γ(t) originating at xx, the point γ(l)γ(l) is ϵϵ-close to xx. We use Low’s notion of refocussing Lorentzian space–times to show that if (Mm,g),m>1(Mm,g),m>1 is a YxYx-manifold, then MM is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a YxYx-manifold is a YxYx-manifold. Another corollary is that if (Mm,g),m=2,3(Mm,g),m=2,3 is a YxYx-manifold, then (M,h)(M,h) is a Ylx-manifold for some metric hh.