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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.05907 |
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| _version_ | 1866909099147395072 |
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| author | Caramello Jr, Francisco C. Martins, Henrique A. Puel Silva, Ivan P. Costa e |
| author_facet | Caramello Jr, Francisco C. Martins, Henrique A. Puel Silva, Ivan P. Costa e |
| contents | We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transverse Geometry of Lorentzian foliations with applications to Lorentzian orbifolds Caramello Jr, Francisco C. Martins, Henrique A. Puel Silva, Ivan P. Costa e Differential Geometry 53C12, 53C50 We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds. |
| title | Transverse Geometry of Lorentzian foliations with applications to Lorentzian orbifolds |
| topic | Differential Geometry 53C12, 53C50 |
| url | https://arxiv.org/abs/2402.05907 |