Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914408997847040 |
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| author | Bauermeister, Friedrich |
| author_facet | Bauermeister, Friedrich |
| contents | 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 $Y^x_l$ 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds Bauermeister, Friedrich Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics Geometric Topology Symplectic Geometry 53C50 (Primary) 53C20, 53C22, 57R17 (Secondary) 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 $Y^x_l$ 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 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. |
| title | Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics Geometric Topology Symplectic Geometry 53C50 (Primary) 53C20, 53C22, 57R17 (Secondary) |
| url | https://arxiv.org/abs/2503.23565 |