Positively curved homogeneous metrics on spheres

TL;DR

This paper classifies all homogeneous metrics with positive sectional curvature on spheres, addressing a complex problem in differential geometry and determining pinching constants where possible.

Contribution

It provides a comprehensive classification of positively curved homogeneous metrics on spheres, resolving a longstanding subtle and difficult problem.

Findings

Complete classification of homogeneous metrics with positive curvature on spheres.

Identification of pinching constants for certain metrics.

Exclusion of one Aloff Wallach space from the classification.

Abstract

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one special Aloff Wallach space where the set of homogeneous metrics is too large).