Seminar on low-dimensional mathematics
June 18, 2010. Room 311, beginning at 16:00.

Vladimir Chernov (Dartmouth, USA).

"Refocussing and $\tilde Y^x$-manifolds." 
Based on the works of Kinlaw, Sadykov, Rudyak and the speaker.

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 $h$.

The notions of $\tilde Y^x$ and $Y^x_l$-manifolds are closely related to
refocusing spacetimes. We discuss the relation and some of the reasons for
why these spacetimes are important.  We also discuss recent results Kinlaw
saying that a Lorentz covering of a refocusing space-time is also
refocusing and that the possibly empty set of points where the refocusing
happens is closed.
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