Betti numbers and the curvature operator of the second kind
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913527914037248 |
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| author | Nienhaus, Jan Petersen, Peter Wink, Matthias |
| author_facet | Nienhaus, Jan Petersen, Peter Wink, Matthias |
| contents | We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number provided that the curvature operator of the second kind is $C(p,n)$-positive. Our curvature conditions become weaker as $p$ increases. For $p=\frac{n}{2}$ we have $C(p,n)= \frac{3n}{2} \frac{n+2}{n+4} $, and for $5 \leq p \leq \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature operator of the second kind with negative Ricci curvatures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14218 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Betti numbers and the curvature operator of the second kind Nienhaus, Jan Petersen, Peter Wink, Matthias Differential Geometry 53B20, 53C20, 53C21, 58A14 We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number provided that the curvature operator of the second kind is $C(p,n)$-positive. Our curvature conditions become weaker as $p$ increases. For $p=\frac{n}{2}$ we have $C(p,n)= \frac{3n}{2} \frac{n+2}{n+4} $, and for $5 \leq p \leq \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature operator of the second kind with negative Ricci curvatures. |
| title | Betti numbers and the curvature operator of the second kind |
| topic | Differential Geometry 53B20, 53C20, 53C21, 58A14 |
| url | https://arxiv.org/abs/2206.14218 |