Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds

Abstract

Let $(M,h)$ be a connected, complete Riemannian manifold, $x\in M$, and $l>0$. Then $M$ is called a $Z^x$ manifold if all geodesics starting at $x$ return to $x$, and it is called a $Y_l^x$ manifold if every unit-speed geodesic starting at $x$ returns to $x$ at time $l$. It is unknown whether there are $Z^x$ manifolds that are not $Y_l^x$ manifolds for any $l>0$. By the Bérard-Bergery theorem, any $Y_l^x$ manifold of dimension at least $2$ is compact with finite fundamental group. We prove the same result for $Z^x$ manifolds $M$ for which all unit-speed geodesics starting at $x$ return to $x$ in uniformly bounded time. We also prove that any $Z^x$ manifold $(M,h)$ with $h$ analytic is a $Y_l^x$ manifold for some $l>0$. We start by defining a class of globally hyperbolic spacetimes (called observer-refocusing) such that any $Z^x$ manifold is the Cauchy surface of some observer-refocusing spacetime. We then prove that under suitable conditions the Cauchy surfaces of observer-refocusing spacetimes are compact with finite fundamental group, and we show that analytic observer-refocusing spacetimes of dimension at least $3$ are strongly refocusing. We end by stating a contact-theoretic conjecture analogous to our results in Riemannian and Lorentzian geometry.

arXiv comment27 pages. version 6 and final version

References in corpus (12)

• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes
• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Non-negative Legendrian isotopy in $S{T}^{\ast }M$
• Linking and causality in globally hyperbolic spacetimes
• Universal orderability of Legendrian isotopy classes
• On the space of light rays of a space-time and a reconstruction theorem by Low
• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Refocusing of Light Rays in Space-Time
• Causality and Legendrian linking for higher dimensional spacetimes
• Topological properties of manifolds admitting a $Y^x$-Riemannian metric