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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.00471 |
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| _version_ | 1866917412417306624 |
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| author | Navarro, Dimitri Pan, Jiayin |
| author_facet | Navarro, Dimitri Pan, Jiayin |
| contents | We prove that a sub-Riemannian manifold equipped with a full-support Radon measure is never $\mathrm{CD}(K,N)$ for any $K\in \mathbb{R}$ and $N\in (1,\infty)$ unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new $\mathrm{RCD}$ structures on $\mathbb{R}^n$, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal non-CD of sub-Riemannian manifolds Navarro, Dimitri Pan, Jiayin Differential Geometry We prove that a sub-Riemannian manifold equipped with a full-support Radon measure is never $\mathrm{CD}(K,N)$ for any $K\in \mathbb{R}$ and $N\in (1,\infty)$ unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new $\mathrm{RCD}$ structures on $\mathbb{R}^n$, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations. |
| title | Universal non-CD of sub-Riemannian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.00471 |