$\delta$-homogeneity in Finsler geometry and the positive curvature problem

TL;DR

This paper investigates $ ext{delta}$-homogeneity in Finsler geometry, showing it closely relates to normal homogeneity, and uses this relationship to classify positively curved $ ext{delta}$-homogeneous Finsler spaces.

Contribution

It establishes that $ ext{delta}$-homogeneous Finsler spaces can be approximated by normal homogeneous ones, enabling classification of positively curved $ ext{delta}$-homogeneous spaces.

Findings

$ ext{delta}$-homogeneous Finsler metrics are limits of normal homogeneous metrics.

Classification of positively curved $ ext{delta}$-homogeneous Finsler spaces matches that of normal homogeneous spaces.

The approximation technique applies to $ ext{delta}$-homogeneous metrics satisfying the (FP) condition.

Abstract

In this paper, we explore the similarity between normal homogeneity and $\delta$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $\delta$-homogeneous Finsler space is $G$-$\delta$-homo-geneous, for some suitably chosen connected quasi-compact $G$. So $\delta$-homogeneous Finsler metrics can be defined by a bi-invariant singular metric on $G$ and submersion, just as normal homogeneous metrics, using a bi-invariant Finsler metric on $G$ instead. More careful analysis shows, in the space of all Finsler metrics on $G/H$, the subset of all $G$-$\delta$-homogeneous ones is in fact the closure for the subset of all $G$-normal ones, in the local $C^0$-topology (Theorem \ref{main-thm-1}). Using this approximation technique, the classification work for positively curved normal homogeneous Finsler spaces can be applied to classify positively curved $\delta$-homogeneous Finsler spaces, which provides the same classification list. As a by-product, this argument tells more about $\delta$-homogeneous Finsler metrics satisfying the (FP) condition (a weaker version of positively curved condition).