New rigidity theorem of Einstein manifolds and curvature operator of the second kind
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918326449471488 |
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| author | Fu, Haiping Lu, Yao |
| author_facet | Fu, Haiping Lu, Yao |
| contents | Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity. Furthermore, employing a result of Li \cite{Li5}, we establish that any closed Einstein manifold of dimension $n \ge 4$ satisfying
\[k^{-1}({λ}_1+\cdots +{λ}_k)\ge -θ(n,k) \bar{λ},\quad \text{for some} \quad k \le [\frac{n+2}{4}]\]
must be either flat or a spherical space form. Here, ${λ}_1\le {λ}_2\le \cdots \le {λ}_{\frac{(n-1)(n+2)}{2}}$ are the eigenvalues of $\mathring{R}\,$, $\bar{λ}$ is their average, and $θ(n,k)$ is a positive constant. This result generalizes the work of Dai-Fu \cite{DF} and Chen-Wang \cite{CW1,CW}.We also classify four-dimensional Einstein manifolds satisfying a cone condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21496 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New rigidity theorem of Einstein manifolds and curvature operator of the second kind Fu, Haiping Lu, Yao Differential Geometry Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity. Furthermore, employing a result of Li \cite{Li5}, we establish that any closed Einstein manifold of dimension $n \ge 4$ satisfying \[k^{-1}({λ}_1+\cdots +{λ}_k)\ge -θ(n,k) \bar{λ},\quad \text{for some} \quad k \le [\frac{n+2}{4}]\] must be either flat or a spherical space form. Here, ${λ}_1\le {λ}_2\le \cdots \le {λ}_{\frac{(n-1)(n+2)}{2}}$ are the eigenvalues of $\mathring{R}\,$, $\bar{λ}$ is their average, and $θ(n,k)$ is a positive constant. This result generalizes the work of Dai-Fu \cite{DF} and Chen-Wang \cite{CW1,CW}.We also classify four-dimensional Einstein manifolds satisfying a cone condition. |
| title | New rigidity theorem of Einstein manifolds and curvature operator of the second kind |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.21496 |