Geometrically formal 4-manifolds with nonnegative sectional curvature

TL;DR

This paper classifies compact 4-manifolds with nonnegative sectional curvature that are geometrically formal, showing they are topologically or diffeomorphically equivalent to well-known spaces like S^4 or CP^2, and extends results related to the Hopf conjecture.

Contribution

It provides a classification of geometrically formal 4-manifolds with nonnegative curvature, including cases with positive curvature and relaxed conditions, advancing understanding of their topology.

Findings

Manifolds with positive sectional curvature are homeomorphic to S^4 or diffeomorphic to CP^2.

The classification extends to manifolds with nearly positive curvature and relaxed harmonic form conditions.

Supports the Hopf conjecture for S^2 x S^2 within this class of manifolds.

Abstract

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to CP^2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S^2 x S^2 holds in this class of manifolds.