Sectional Curvature, Isotropic Curvature, and Yau’s Pinching Problem
Abstract
Abstract We prove that if a closed Riemannian manifold $$(M^n,g)$$ ( M n , g ) has finite fundamental group and satisfies the curvature condition $$\begin{aligned} R_{1313} +R_{1414} +R_{2323} + R_{2424} > \tfrac{1}{2}\left( R_{1212} + R_{3434}\right) \end{aligned}$$ R 1313 + R 1414 + R 2323 + R 2424 > 1 2 R 1212 + R 3434 for all orthonormal four-frame $$\{e_1, e_2, e_3, e_4\} \subset T_pM$$ { e 1 , e 2 , e 3 , e 4 } ⊂ T p M , then the universal cover of M is homeomorphic to the n -sphere. This generalizes the famous sphere theorem under the stronger condition of $$\frac{1}{4}$$ 1 4 -pinched sectional curvature. As an application, we provide a partial answer to a pinching problem proposed by Yau in 1990.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Holomorphic and Operator Theory
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Journal of Geometric Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 36
- Issue
- 8
- Publication date
- 10 July 2026
- Language
- EN