Curvature of the Second kind and a conjecture of Nishikawa
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866909997144735744 |
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| author | Gursky, Matthew Cao, Xiaodong Tran, Hung |
| author_facet | Gursky, Matthew Cao, Xiaodong Tran, Hung |
| contents | In this paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the curvature (operator) of the second kind are positive are diffeomorphic to a sphere, by showing that such manifolds satisfy Brendle's PIC1 condition. In dimension four we show that curvature of the second kind has a canonical normal form, and use this to classify Einstein four-manifolds for which the curvature (operator) of the second kind is five-non-negative. We also calculate the normal form for some explicit examples in order to show that this assumption is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_01212 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Curvature of the Second kind and a conjecture of Nishikawa Gursky, Matthew Cao, Xiaodong Tran, Hung Differential Geometry Analysis of PDEs 53C21, 53C24, 53C25 In this paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the curvature (operator) of the second kind are positive are diffeomorphic to a sphere, by showing that such manifolds satisfy Brendle's PIC1 condition. In dimension four we show that curvature of the second kind has a canonical normal form, and use this to classify Einstein four-manifolds for which the curvature (operator) of the second kind is five-non-negative. We also calculate the normal form for some explicit examples in order to show that this assumption is sharp. |
| title | Curvature of the Second kind and a conjecture of Nishikawa |
| topic | Differential Geometry Analysis of PDEs 53C21, 53C24, 53C25 |
| url | https://arxiv.org/abs/2112.01212 |