Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866909226231660544 |
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| author | Tashiro, Kenshiro |
| author_facet | Tashiro, Kenshiro |
| contents | In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the Popp's volume or the minimal Popp's volume converges to zero. In the appendix, we give the systolic inequality on sub-Riemannian Heisenberg manifolds, and observe that the exponent of the total measure is equal to the inverse of the Hausdorff dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_06806 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics Tashiro, Kenshiro Differential Geometry 53C17, 28A78, 20F18 In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the Popp's volume or the minimal Popp's volume converges to zero. In the appendix, we give the systolic inequality on sub-Riemannian Heisenberg manifolds, and observe that the exponent of the total measure is equal to the inverse of the Hausdorff dimension. |
| title | Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics |
| topic | Differential Geometry 53C17, 28A78, 20F18 |
| url | https://arxiv.org/abs/2010.06806 |