Geometrically formal homogeneous metrics of positive curvature

TL;DR

This paper investigates the special class of geometrically formal homogeneous metrics with positive curvature, showing they are either symmetric or on rational homology spheres, linking geometric formality with topological properties.

Contribution

It characterizes geometrically formal homogeneous metrics of positive curvature, revealing they are either symmetric or on rational homology spheres, advancing understanding of curvature and topology.

Findings

Geometrically formal homogeneous metrics are either symmetric or on rational homology spheres.

Such metrics of positive curvature are topologically constrained.

The work connects geometric formality with rational homotopy theory.

Abstract

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.