New Sphere Theorems under Curvature Operator of the Second Kind
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917046446456832 |
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| author | Li, Xiaolong |
| author_facet | Li, Xiaolong |
| contents | We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} α^{-1} (λ_1 +\cdots +λ_α) > - θ\barλ, \end{equation*} where $λ_1 \leq \cdots \leq λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\barλ$ is their average, and $θ> -1$. Under such conditions with optimal $θ$ depending on $n$ and $α$, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to $θ=0$ of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New Sphere Theorems under Curvature Operator of the Second Kind Li, Xiaolong Differential Geometry 53C21, 53C24, 53C20, 53C55 We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} α^{-1} (λ_1 +\cdots +λ_α) > - θ\barλ, \end{equation*} where $λ_1 \leq \cdots \leq λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\barλ$ is their average, and $θ> -1$. Under such conditions with optimal $θ$ depending on $n$ and $α$, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to $θ=0$ of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results. |
| title | New Sphere Theorems under Curvature Operator of the Second Kind |
| topic | Differential Geometry 53C21, 53C24, 53C20, 53C55 |
| url | https://arxiv.org/abs/2407.13847 |