Einstein manifolds under cone conditions for the curvature operator of the second kind
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909737529901056 |
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| author | Cheng, Haiqing Wang, Kui |
| author_facet | Cheng, Haiqing Wang, Kui |
| contents | It is established in [6, 14, 23] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [18]. Precisely, we prove that any closed Einstein manifold of dimension $n=4$ or $n=5$ or $n\ge 8$, if the curvature operator of the second kind $\mathring{R}$ satisfies \begin{align*}
(λ_1+λ_2)/2 \ge -θ(n) \bar λ, \end{align*} then the manifold is either flat or a round sphere. Here, $λ_1\le λ_2\le \cdots\le λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $ \bar λ$ is their average, and $θ(n)$ is a positive constant defined as in (1.2). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_11226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Einstein manifolds under cone conditions for the curvature operator of the second kind Cheng, Haiqing Wang, Kui Differential Geometry It is established in [6, 14, 23] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [18]. Precisely, we prove that any closed Einstein manifold of dimension $n=4$ or $n=5$ or $n\ge 8$, if the curvature operator of the second kind $\mathring{R}$ satisfies \begin{align*} (λ_1+λ_2)/2 \ge -θ(n) \bar λ, \end{align*} then the manifold is either flat or a round sphere. Here, $λ_1\le λ_2\le \cdots\le λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $ \bar λ$ is their average, and $θ(n)$ is a positive constant defined as in (1.2). |
| title | Einstein manifolds under cone conditions for the curvature operator of the second kind |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.11226 |