Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem

Abstract: 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.