Manifolds with nonnegative curvature operator of the second kind

TL;DR

This paper classifies Riemannian manifolds with certain positivity conditions on the curvature operator of the second kind, showing they are diffeomorphic to well-understood geometric models, thus advancing understanding of curvature constraints.

Contribution

It proves new classification results for manifolds with three-positive or three-nonnegative curvature operator of the second kind, refining previous assumptions and settling parts of Nishikawa's conjecture.

Findings

Manifolds with three-positive curvature operator are diffeomorphic to spherical space forms.

Manifolds with three-nonnegative curvature operator are either spherical space forms, flat, or quotients of symmetric spaces.

The results improve previous classifications by weakening positivity assumptions.

Abstract

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa's conjecture under a weaker assumption.