On the topology of manifolds with nonnegative Ricci curvature and linear volume growth
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913557021458432 |
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| author | Navarro, Dimitri Pan, Jiayin Zhu, Xingyu |
| author_facet | Navarro, Dimitri Pan, Jiayin Zhu, Xingyu |
| contents | Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the topology of manifolds with nonnegative Ricci curvature and linear volume growth Navarro, Dimitri Pan, Jiayin Zhu, Xingyu Differential Geometry Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces. |
| title | On the topology of manifolds with nonnegative Ricci curvature and linear volume growth |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.15488 |