Local rigidity of group actions of isometries on compact Riemannian manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909605932564480 |
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| author | Stolovitch, Laurent Zhao, Zhiyan |
| author_facet | Stolovitch, Laurent Zhao, Zhiyan |
| contents | In this article, we consider perturbations of isometries on a compact Riemannian manifold $M$. We investigate the smooth (resp. analytic) rigidity phenomenon of groups of these isometries. As a particular case, we prove that if a finite family of smooth (resp. analytic) small enough perturbations is simultaneously conjugate to the family of isometries via a finitely smooth diffeomorphism, then it is simultaneously smoothly (resp. analytically) conjugate to it whenever the family of isometries satisfies a Diophantine condition. Our results generalize the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations as well as Fisher-Margulis's theorem on group actions satisfying Kazhdan's property (T). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local rigidity of group actions of isometries on compact Riemannian manifolds Stolovitch, Laurent Zhao, Zhiyan Dynamical Systems In this article, we consider perturbations of isometries on a compact Riemannian manifold $M$. We investigate the smooth (resp. analytic) rigidity phenomenon of groups of these isometries. As a particular case, we prove that if a finite family of smooth (resp. analytic) small enough perturbations is simultaneously conjugate to the family of isometries via a finitely smooth diffeomorphism, then it is simultaneously smoothly (resp. analytically) conjugate to it whenever the family of isometries satisfies a Diophantine condition. Our results generalize the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations as well as Fisher-Margulis's theorem on group actions satisfying Kazhdan's property (T). |
| title | Local rigidity of group actions of isometries on compact Riemannian manifolds |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2505.05884 |