Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915392539066368 |
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| author | Kunzinger, Michael Ohanyan, Argam Vardabasso, Alessio |
| author_facet | Kunzinger, Michael Ohanyan, Argam Vardabasso, Alessio |
| contents | We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $C^1$. With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $\mathsf{RCD}$-splitting theorem \cite{gigli2013splitting, gigli2014splitoverview}. Along the way, we establish a Bochner-Weitzenböck identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \cite{mondino2024equivalence}. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see \cite{mondino2024equivalence}) between distributional Ricci curvature bounds and $\mathsf{RCD}$-type bounds, using in part the stability of the variable $\mathsf{CD}$-condition under suitable limits \cite{ketterer2017variableCD}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics Kunzinger, Michael Ohanyan, Argam Vardabasso, Alessio Differential Geometry Metric Geometry 53C21, 53C24, 46T30, 49Q22 We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $C^1$. With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $\mathsf{RCD}$-splitting theorem \cite{gigli2013splitting, gigli2014splitoverview}. Along the way, we establish a Bochner-Weitzenböck identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \cite{mondino2024equivalence}. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see \cite{mondino2024equivalence}) between distributional Ricci curvature bounds and $\mathsf{RCD}$-type bounds, using in part the stability of the variable $\mathsf{CD}$-condition under suitable limits \cite{ketterer2017variableCD}. |
| title | Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics |
| topic | Differential Geometry Metric Geometry 53C21, 53C24, 46T30, 49Q22 |
| url | https://arxiv.org/abs/2406.06762 |