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

Friedrich Bauermeister

TL;DR

The paper connects geodesic return properties in Riemannian manifolds with the topology of the underlying space through Lorentzian observer-refocusing spacetimes. It proves that if all unit-speed geodesics from a point $x$ return to a point $y$ within a uniform time, the manifold is compact with finite fundamental group, and in the analytic setting this implies the stronger $Y^{(x,y)}_l$ property. By translating to globally hyperbolic spacetimes and employing observer-refocusing concepts, the authors extend Bott-Samelson-type conclusions to Lorentzian and analytic contexts and demonstrate strong refocusing under analyticity. They also establish a bridge to contact geometry via positive Legendrian isotopies and Reeb flows, culminating in conjectures that generalize the results to the contact setting and highlighting potential topological constraints induced by refocusing dynamics.

Abstract

Let $(M,h)$ be a connected, complete Riemannian manifold, let $x\in M$ 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 $Y^x_l$ 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 $Y^x_l$ manifolds for any $l>0$. By the Bérard-Bergery theorem, any $Y^x_l$ 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 $Y^x_l$ 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 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.

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

TL;DR

The paper connects geodesic return properties in Riemannian manifolds with the topology of the underlying space through Lorentzian observer-refocusing spacetimes. It proves that if all unit-speed geodesics from a point return to a point within a uniform time, the manifold is compact with finite fundamental group, and in the analytic setting this implies the stronger property. By translating to globally hyperbolic spacetimes and employing observer-refocusing concepts, the authors extend Bott-Samelson-type conclusions to Lorentzian and analytic contexts and demonstrate strong refocusing under analyticity. They also establish a bridge to contact geometry via positive Legendrian isotopies and Reeb flows, culminating in conjectures that generalize the results to the contact setting and highlighting potential topological constraints induced by refocusing dynamics.

Abstract

Let be a connected, complete Riemannian manifold, let and . Then is called a manifold if all geodesics starting at return to and it is called a manifold if every unit-speed geodesic starting at returns to at time . It is unknown whether there are manifolds that are not manifolds for any . By the Bérard-Bergery theorem, any manifold of dimension at least is compact with finite fundamental group. We prove the same result for manifolds for which all unit-speed geodesics starting at return to in uniformly bounded time. We also prove that any manifold with analytic is a manifold for some . We start by defining a class of globally hyperbolic spacetimes (called observer-refocusing) such that any 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 show that analytic observer-refocusing spacetimes of dimension at least are strongly refocusing. We end by stating a contact-theoretic conjecture analogous to our results in Riemannian and Lorentzian geometry.

Paper Structure

This paper contains 6 sections, 26 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Related results about $Y^x_l$ manifolds and refocusing spacetimes
  4. Observer-Refocusing Spacetimes
  5. The Analytic Case
  6. Analogous conjecture in the Contact case

Key Result

Proposition 1.9

If $(M,h)$ is a $Y^{(x,y)}_l$ manifold, then the associated spacetime $(X,g)$ is a strongly refocusing spacetime with $p=(x,0)$ and $q=(y,l)$. If $(M,h)$ is a $Z^{(x,y)}$ manifold, then the associated spacetime $(X,g)$ is an observer-refocusing spacetime with $p=(x,0)$ and observer $\gamma:\mathbb{R

Theorems & Definitions (75)

  • Definition 1.1: $Y^{(x,y)}_l$ manifold
  • Definition 1.2: $Z^{(x,y)}$ manifold
  • Definition 1.3: strongly refocusing spacetime
  • Definition 1.4: observer-refocusing spacetime
  • Definition 1.5: spacetime associated to a Riemannian manifold
  • Definition 1.6: globally hyperbolic spacetime
  • Definition 1.7: temporal functions
  • Remark 1.8
  • Proposition 1.9
  • proof
  • ...and 65 more