Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909760188579840 |
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| author | Li, Xiaolong |
| author_facet | Li, Xiaolong |
| contents | We prove that if a closed Riemannian manifold $(M^n,g)$ has finite fundamental group and satisfies the curvature condition \begin{equation*}
R_{1313} +R_{1414} +R_{2323} + R_{2424} > \tfrac{1}{2}\left(R_{1212} + R_{3434}\right) \end{equation*} for all orthonormal four-frame $\{e_1, e_2, e_3, e_4\} \subset T_pM$, then $M$ is homeomorphic to a spherical space form. This generalizes the famous sphere theorem under the stronger condition of $\frac{1}{4}$-pinched sectional curvature. As an application, we provide a partial answer to a pinching problem proposed by Yau in 1990. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem Li, Xiaolong Differential Geometry 53C20, 53C21 We prove that if a closed Riemannian manifold $(M^n,g)$ has finite fundamental group and satisfies the curvature condition \begin{equation*} R_{1313} +R_{1414} +R_{2323} + R_{2424} > \tfrac{1}{2}\left(R_{1212} + R_{3434}\right) \end{equation*} for all orthonormal four-frame $\{e_1, e_2, e_3, e_4\} \subset T_pM$, then $M$ is homeomorphic to a spherical space form. This generalizes the famous sphere theorem under the stronger condition of $\frac{1}{4}$-pinched sectional curvature. As an application, we provide a partial answer to a pinching problem proposed by Yau in 1990. |
| title | Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem |
| topic | Differential Geometry 53C20, 53C21 |
| url | https://arxiv.org/abs/2508.21661 |