Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866914245349736448 |
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| author | Cheng, Haiqing Wang, Kui |
| author_facet | Cheng, Haiqing Wang, Kui |
| contents | In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ α^{-1}\big(\sum_{i=1}^{[α]} λ_i+ (α- [α] ) λ_{[α] + 1} \big) \ge -θ\barλ \] for some real number $α\in [1, (n+2)(n-1)/2)$. Here $[α] :=\max\{ m \in \mathbb{Z}: m \leq α\}$, $θ>-1$ and $λ_1 \le \cdots \le λ_{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\barλ$ is their average. The main result states that any closed Einstein manifold of dimension $n \ge 4$ with $\mathring{R}$ satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to $α\in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06556 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds Cheng, Haiqing Wang, Kui Differential Geometry 53C20, 53C24, 53C25 In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ α^{-1}\big(\sum_{i=1}^{[α]} λ_i+ (α- [α] ) λ_{[α] + 1} \big) \ge -θ\barλ \] for some real number $α\in [1, (n+2)(n-1)/2)$. Here $[α] :=\max\{ m \in \mathbb{Z}: m \leq α\}$, $θ>-1$ and $λ_1 \le \cdots \le λ_{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\barλ$ is their average. The main result states that any closed Einstein manifold of dimension $n \ge 4$ with $\mathring{R}$ satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to $α\in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}. |
| title | Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds |
| topic | Differential Geometry 53C20, 53C24, 53C25 |
| url | https://arxiv.org/abs/2601.06556 |