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Topological properties of manifolds admitting a $Y^x$-Riemannian metric

A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every unit speed geodesic $γ(t)$ originating at $γ(0)=x\in M$ satisfies $γ(l)=x$ for $0\neq l\in \R$. Bérard-Bergery proved that if $(M^m,g), m>1$ is a $Y^x_l$-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 $ε>0$ there exists $l>ε$ such that for every unit speed geodesic $γ(t)$ originating at $x$, the point $γ(l)$ is $ε$-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 $Y^x_l$-manifold for some metric $h.$