Manifolds with 1/4-pinched flag curvature

TL;DR

This paper introduces the concept of quarter pinched flag curvature in nonnegatively curved manifolds and proves that such manifolds have nonnegative complex sectional curvature, leading to classification results for positively curved cases.

Contribution

It establishes that manifolds with quarter pinched flag curvature have nonnegative complex sectional curvature and classifies positively curved manifolds under this condition, extending previous results.

Findings

Manifolds with quarter pinched flag curvature have nonnegative complex sectional curvature.

Positively curved manifolds with strictly quarter pinched flag curvature are space forms.

Results extend to odd-dimensional manifolds with pinching constants below one quarter.

Abstract

We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain results for the case that the flag curvature is pinched with some constant below one quarter, one of which generalizes a recent work of Petersen and Tao.