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

TL;DR

This paper explores the topological constraints of manifolds admitting special Riemannian metrics related to geodesic properties, extending previous results and linking Lorentzian refocussing to Riemannian geometry.

Contribution

It introduces the concept of $Y^x$-manifolds, generalizes known results about $Y^x_l$-manifolds, and connects Lorentzian refocussing to Riemannian topology.

Findings

$Y^x$-manifolds are closed with finite fundamental group.

Coverings of $Y^x$-manifolds are also $Y^x$-manifolds.

In dimensions 2 and 3, $Y^x$-manifolds are equivalent to $Y^x_l$-manifolds.

Abstract

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 $h.$