A local Lorentzian Ferrand-Obata theorem for conformal vector fields
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908631382884352 |
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| author | Dumitrescu, Sorin Frances, Charles Melnick, Karin Pecastaing, Vincent Zeghib, Abdelghani |
| author_facet | Dumitrescu, Sorin Frances, Charles Melnick, Karin Pecastaing, Vincent Zeghib, Abdelghani |
| contents | For a conformal vector field on a closed, real-analytic, Lorentzian manifold we prove that the flow is locally isometric -- that it preserves a metric in the conformal class on a neighborhood of any point -- or the metric is everywhere conformally flat. The main theorem can be viewed as a local version of the Lorentzian Lichnerowicz conjecture in the real-analytic setting. The key result is an optimal improvement of the local normal forms for conformal vector fields of [FM13], which focused on non-linearizable singularities. This article is primarily concerned with essential linearizable singularities, and the proofs include global arguments which rely on the compactness assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A local Lorentzian Ferrand-Obata theorem for conformal vector fields Dumitrescu, Sorin Frances, Charles Melnick, Karin Pecastaing, Vincent Zeghib, Abdelghani Differential Geometry Dynamical Systems 53B30, 53C24, 53A30 For a conformal vector field on a closed, real-analytic, Lorentzian manifold we prove that the flow is locally isometric -- that it preserves a metric in the conformal class on a neighborhood of any point -- or the metric is everywhere conformally flat. The main theorem can be viewed as a local version of the Lorentzian Lichnerowicz conjecture in the real-analytic setting. The key result is an optimal improvement of the local normal forms for conformal vector fields of [FM13], which focused on non-linearizable singularities. This article is primarily concerned with essential linearizable singularities, and the proofs include global arguments which rely on the compactness assumption. |
| title | A local Lorentzian Ferrand-Obata theorem for conformal vector fields |
| topic | Differential Geometry Dynamical Systems 53B30, 53C24, 53A30 |
| url | https://arxiv.org/abs/2511.03713 |