  A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every unit
speed geodesic $\gamma(t)$ originating at $\gamma(0)=x\in M$ satisfies
$\gamma(l)=x$ for $0\neq l\in \R$. B\'erard-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 $\epsilon >0$ there
exists $l>\epsilon$ such that for every unit speed geodesic $\gamma(t)$
originating at $x$, the point $\gamma(l)$ is $\epsilon$-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 $h.$
