New Curvature Conditions for the Bochner Technique
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866913527871045632 |
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| author | Petersen, Peter Wink, Matthias |
| author_facet | Petersen, Peter Wink, Matthias |
| contents | We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5,6$ we obtain vanishing of the Betti numbers provided that the curvature operator is $3$-positive. As Böhm-Wilking observed, $3$-positivity of the curvature operator is not preserved by the Ricci flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_09958 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | New Curvature Conditions for the Bochner Technique Petersen, Peter Wink, Matthias Differential Geometry 53B20, 53C20, 53C21, 53C23, 58A14 We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5,6$ we obtain vanishing of the Betti numbers provided that the curvature operator is $3$-positive. As Böhm-Wilking observed, $3$-positivity of the curvature operator is not preserved by the Ricci flow. |
| title | New Curvature Conditions for the Bochner Technique |
| topic | Differential Geometry 53B20, 53C20, 53C21, 53C23, 58A14 |
| url | https://arxiv.org/abs/1908.09958 |