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

Abstract

A complete Riemannian manifold $(M,g)$ is a $Y_l^x$-manifold if every unit speed geodesic $\gamma (t)$ originating at $\gamma (0)=x\in M$ satisfies $\gamma (l)=x$ for $0\ne l\in \mathbb{R}$. Bérard-Bergery proved that if $(M^m,g),m>1$ is a $Y_l^x$-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 $Y_l^x$-manifold for some metric $h.$

14 pages

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Cited by in corpus (3)

• Refocusing of Light Rays in Space-Time
• Conjectures on the Relations of Linking and Causality in Causally Simple Spacetimes
• Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds