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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.15576 |
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| _version_ | 1866911486991925248 |
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| author | Li, Xiang-Dong |
| author_facet | Li, Xiang-Dong |
| contents | In this paper, we use the information-theoretic approach to study curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds. We prove the equivalence of the ${\rm CD}(K, m)$-condition for $K\in \mathbb{R}$ and $m\in [n, \infty]$ and a family of Shannon and Rényi entropy differential inequalities along the geodesics on the Wasserstein space over a Riemannian manifold. {The rigidity models of the enhanced entropy differential inequalities are the $K$-Einstein manifolds and the $(K, m)$-Einstein manifolds}. Moreover, we prove the monotonicity and rigidity theorem of the $W$-entropy associated with the Shannon entropy and the Rényi entropy along the geodesics on the Wasserstein space over Riemannian manifolds with CD$(0, m)$-condition. Comparing with the characterization of the the CD$(K, m)$ curvature-dimension condition in the framework of the synthetic geometry developed by Lott, Sturm and Villani, we provide more simple equivalent characterizations for the CD$(K, m)$-condition, and we provide a characterization of the Einstein and quasi-Einstein manifolds by the enhanced entropy differential equality and the enhanced entropy power differential equality. These are new in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds Li, Xiang-Dong Differential Geometry In this paper, we use the information-theoretic approach to study curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds. We prove the equivalence of the ${\rm CD}(K, m)$-condition for $K\in \mathbb{R}$ and $m\in [n, \infty]$ and a family of Shannon and Rényi entropy differential inequalities along the geodesics on the Wasserstein space over a Riemannian manifold. {The rigidity models of the enhanced entropy differential inequalities are the $K$-Einstein manifolds and the $(K, m)$-Einstein manifolds}. Moreover, we prove the monotonicity and rigidity theorem of the $W$-entropy associated with the Shannon entropy and the Rényi entropy along the geodesics on the Wasserstein space over Riemannian manifolds with CD$(0, m)$-condition. Comparing with the characterization of the the CD$(K, m)$ curvature-dimension condition in the framework of the synthetic geometry developed by Lott, Sturm and Villani, we provide more simple equivalent characterizations for the CD$(K, m)$-condition, and we provide a characterization of the Einstein and quasi-Einstein manifolds by the enhanced entropy differential equality and the enhanced entropy power differential equality. These are new in the literature. |
| title | Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2407.15576 |