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

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

2010-05-27

arxiv.org/abs/1005.5075v1

Journal of Geometry and Physics (60) 2010 1530-1538

Mathematics

Differential Geometry

14 pages

Scientific paper

10.1016/j.geomphys.2010.05.010

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 $h.$

Affiliated with

Also associated with

No associations

Rating

Topological properties of manifolds admitting a $Y^x$-Riemannian metric does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Topological properties of manifolds admitting a $Y^x$-Riemannian metric, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Topological properties of manifolds admitting a $Y^x$-Riemannian metric will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-192483