Reversible homogeneous Finsler metrics with positive flag curvature

TL;DR

This paper classifies homogeneous reversible Finsler metrics with positive flag curvature, showing most admit Riemannian metrics with positive sectional curvature, except for a few low-dimensional cases.

Contribution

It provides a classification of such Finsler metrics and links their existence to Riemannian metrics with positive curvature, highlighting unresolved cases.

Findings

Most homogeneous reversible Finsler metrics with positive flag curvature admit Riemannian counterparts.

Identifies specific low-dimensional spaces where the existence of such Finsler metrics remains unknown.

Establishes a connection between Finsler and Riemannian positive curvature conditions.

Abstract

We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions, we do not know if they admit homogeneous Finsler metrics with positive Flag curvature.