Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912459682480128 |
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| author | Pan, Jiayin Ye, Zhu |
| author_facet | Pan, Jiayin Ye, Zhu |
| contents | We study the rigidity problems for open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $M$ properly contains a Euclidean $\mathbb{R}^{k-1}$, then the first Betti number of $M$ is at most $n-k$; moreover, if equality holds, then $M$ is flat. Next, we study the geometry of the orbit $Γ\tilde{p}$, where $Γ=π_1(M,p)$ acts on the universal cover $(\widetilde{M},\tilde{p})$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $Γ\tilde{p}$. We also give the first example of a manifold $M$ of $\mathrm{Ric}>0$ and $π_1(M)=\mathbb{Z}$ but with a varying orbit growth order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity Pan, Jiayin Ye, Zhu Differential Geometry We study the rigidity problems for open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $M$ properly contains a Euclidean $\mathbb{R}^{k-1}$, then the first Betti number of $M$ is at most $n-k$; moreover, if equality holds, then $M$ is flat. Next, we study the geometry of the orbit $Γ\tilde{p}$, where $Γ=π_1(M,p)$ acts on the universal cover $(\widetilde{M},\tilde{p})$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $Γ\tilde{p}$. We also give the first example of a manifold $M$ of $\mathrm{Ric}>0$ and $π_1(M)=\mathbb{Z}$ but with a varying orbit growth order. |
| title | Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2404.10145 |