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Very standard homogeneous Finsler manifolds with positive flag curvature

Xiyun Xu, Ming Xu

TL;DR

The paper proves that for a coset $G/H$ with $G$ a compact connected simply connected simple Lie group and $H$ connected, the existence of positively curved very standard homogeneous Finsler metrics is equivalent to the existence of positively curved homogeneous Riemannian metrics on $G/H$. This yields a complete classification of such Finsler spaces, matching known results from the Riemannian setting and extending Berger-type classifications to the Finsler context. The main strategy combines even-dimensional classifications, reversibility-based obstructions, and Finsler submersion techniques to connect non-reversible odd-dimensional cases to the established Riemannian picture. The resulting list includes compact rank-one symmetric spaces, Berger spaces, Wallach spaces, Aloff–Wallach spaces, and related complex/quaternionic projective examples, confirming that very standard metrics capture all positively curved Finsler geometries in this setting.

Abstract

In this paper, we consider a homogeneous manifold $G/H$ in which $G$ is a compact connected simply connected simple Lie group and $H$ is a closed connected subgroup of $G$. We define standard and very standard homogeneous Finsler metrics on $G/H$, which generalize the standard homogeneous $(α_1,α_2)$ metric in literature. We classify all these $G/H$ which admit positively curved very standard homogeneous Finsler metrics.

Very standard homogeneous Finsler manifolds with positive flag curvature

TL;DR

The paper proves that for a coset with a compact connected simply connected simple Lie group and connected, the existence of positively curved very standard homogeneous Finsler metrics is equivalent to the existence of positively curved homogeneous Riemannian metrics on . This yields a complete classification of such Finsler spaces, matching known results from the Riemannian setting and extending Berger-type classifications to the Finsler context. The main strategy combines even-dimensional classifications, reversibility-based obstructions, and Finsler submersion techniques to connect non-reversible odd-dimensional cases to the established Riemannian picture. The resulting list includes compact rank-one symmetric spaces, Berger spaces, Wallach spaces, Aloff–Wallach spaces, and related complex/quaternionic projective examples, confirming that very standard metrics capture all positively curved Finsler geometries in this setting.

Abstract

In this paper, we consider a homogeneous manifold in which is a compact connected simply connected simple Lie group and is a closed connected subgroup of . We define standard and very standard homogeneous Finsler metrics on , which generalize the standard homogeneous metric in literature. We classify all these which admit positively curved very standard homogeneous Finsler metrics.
Paper Structure (10 sections, 10 theorems, 20 equations)

This paper contains 10 sections, 10 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minkowski norm and Finsler metric
  4. Riemann curvature and flag curvature
  5. Homogeneous Finsler manifold and homogeneous curvature formula
  6. Very standard homogeneous Finsler metric
  7. Proof of Theorem \ref{['main-thm']}
  8. Theme of the proof
  9. The reversible case
  10. The non-reversible case

Key Result

Theorem 1.1

Let $G/H$ be a smooth coset space in which $G$ is a compact connected simply connected simple Lie group, and $H$ is a closed connected subgroup. Then $G/H$ admits very standard homogeneous Finsler metrics with positive flag curvature if and only if it admits positively curved homogeneous Riemannian

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • ...and 3 more