Geometric formality and non-negative scalar curvature

TL;DR

This paper classifies low-dimensional manifolds that admit both non-negative scalar curvature metrics and special harmonic form metrics, extending to higher dimensions under certain Betti number conditions.

Contribution

It provides a classification of manifolds with combined geometric and harmonic form properties, especially in low dimensions and for large first Betti numbers.

Findings

Classifies small-dimensional manifolds with specified curvature and harmonic form conditions.

Extends classification to higher dimensions for manifolds with large first Betti numbers.

Identifies geometric constraints linking scalar curvature and harmonic forms.

Abstract

We classify manifolds of small dimension that admit both, a Riemannian metric of non-negative scalar curvature, and a -- a priori different -- metric for which all wedge products of harmonic forms are harmonic. For manifolds whose first Betti numbers are sufficiently large, this classification extends to higher dimensions.