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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2410.18846 |
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| _version_ | 1866916588554289152 |
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| author | DeVito, Jason Domínguez-Vázquez, Miguel González-Álvaro, David Rodríguez-Vázquez, Alberto |
| author_facet | DeVito, Jason Domínguez-Vázquez, Miguel González-Álvaro, David Rodríguez-Vázquez, Alberto |
| contents | We construct metrics of positive $2^{\rm nd}$ intermediate Ricci curvature, $\mathrm{Ric}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including $\mathbb{S}^6\times\mathbb{S}^7$, $\mathbb{S}^7\times\mathbb{S}^7$ and all their simply connected isometric quotients. In particular, we obtain infinitely many examples in dimension 13. We also produce infinitely many non-simply connected spaces with $\mathrm{Ric}_2>0$ in dimensions 13 and 14, including $\mathbb{RP}^6\times \mathbb{RP}^7$ and $\mathbb{RP}^7\times \mathbb{RP}^7$, which cannot admit a metric of positive sectional curvature. The main new idea is a generalization of the concept of fatness which ensures the existence of $\mathrm{Ric}_2>0$ metrics on the total space of certain homogeneous bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive $\mathrm{Ric}_2$ curvature on products of spheres and their quotients via intermediate fatness DeVito, Jason Domínguez-Vázquez, Miguel González-Álvaro, David Rodríguez-Vázquez, Alberto Differential Geometry We construct metrics of positive $2^{\rm nd}$ intermediate Ricci curvature, $\mathrm{Ric}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including $\mathbb{S}^6\times\mathbb{S}^7$, $\mathbb{S}^7\times\mathbb{S}^7$ and all their simply connected isometric quotients. In particular, we obtain infinitely many examples in dimension 13. We also produce infinitely many non-simply connected spaces with $\mathrm{Ric}_2>0$ in dimensions 13 and 14, including $\mathbb{RP}^6\times \mathbb{RP}^7$ and $\mathbb{RP}^7\times \mathbb{RP}^7$, which cannot admit a metric of positive sectional curvature. The main new idea is a generalization of the concept of fatness which ensures the existence of $\mathrm{Ric}_2>0$ metrics on the total space of certain homogeneous bundles. |
| title | Positive $\mathrm{Ric}_2$ curvature on products of spheres and their quotients via intermediate fatness |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.18846 |