Manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind

TL;DR

This paper proves that four-manifolds with a specific positive curvature condition are diffeomorphic to spherical space forms, and extends some results to higher dimensions, linking curvature conditions to topological classifications.

Contribution

The paper establishes sharp curvature conditions that classify four-manifolds as spherical space forms and extends these results to higher dimensions, connecting curvature positivity to topology.

Findings

Four-manifolds with 4.5-positive curvature operator are diffeomorphic to spherical space forms.

Higher-dimensional manifolds with the same curvature condition are homeomorphic to spherical space forms.

The curvature condition implies positive isotropic and Ricci curvature.

Abstract

We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times \mathbb{S}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\geq 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for $4\frac{1}{2}$-nonnegative curvature operator of the second kind are also obtained.