Sphere Theorems for $\frac{10}{3}$-Positive Curvature Operator of the Second Kind

Abstract

In this paper, we prove that every connected closed Riemannian manifold of dimension $n\geq4$ with $\frac{10}{3}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. We also obtain a classification of connected closed Riemannian manifolds with $\frac{10}{3}$-nonnegative curvature operator of the second kind. The key step is to show that $\frac{10}{3}$-positivity implies the PIC1 condition. Finally, we give an example showing that $\frac{10}{3}$ is optimal for deducing PIC1 from $α$-positivity.