Holonomy restrictions from the curvature operator of the second kind
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910629593350144 |
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| author | Nienhaus, Jan Petersen, Peter Wink, Matthias Wylie, William |
| author_facet | Nienhaus, Jan Petersen, Peter Wink, Matthias Wylie, William |
| contents | We show that an $n$-dimensional Riemannian manifold with $n$-nonnegative or $n$-nonpositive curvature operator of the second kind has restricted holonomy $SO(n)$ or is flat. The result does not depend on completeness and can be improved provided the space is Einstein or Kähler. In particular, if a locally symmetric space has $n$-nonnegative or $n$-nonpositive curvature operator of the second kind, then it has constant curvature. When the locally symmetric space is irreducible this can be improved to $\frac{3n}{2}\frac{n+2}{n+4}$-nonnegative or $\frac{3n}{2}\frac{n+2}{n+4}$-nonpositive curvature operator of the second kind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13820 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Holonomy restrictions from the curvature operator of the second kind Nienhaus, Jan Petersen, Peter Wink, Matthias Wylie, William Differential Geometry 53B20, 53B35, 53C29 We show that an $n$-dimensional Riemannian manifold with $n$-nonnegative or $n$-nonpositive curvature operator of the second kind has restricted holonomy $SO(n)$ or is flat. The result does not depend on completeness and can be improved provided the space is Einstein or Kähler. In particular, if a locally symmetric space has $n$-nonnegative or $n$-nonpositive curvature operator of the second kind, then it has constant curvature. When the locally symmetric space is irreducible this can be improved to $\frac{3n}{2}\frac{n+2}{n+4}$-nonnegative or $\frac{3n}{2}\frac{n+2}{n+4}$-nonpositive curvature operator of the second kind. |
| title | Holonomy restrictions from the curvature operator of the second kind |
| topic | Differential Geometry 53B20, 53B35, 53C29 |
| url | https://arxiv.org/abs/2208.13820 |