Osaka Journal of Mathematics

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

Ming Xu and Lei Zhang*

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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 $G$-$\delta$-homogeneous, 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 1.1). 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).

Article information

Osaka J. Math., Volume 55, Number 1 (2018), 177-194.

First available in Project Euclid: 11 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E46: Semisimple Lie groups and their representations 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]


Xu, Ming; Zhang*, Lei. $\delta$-homogeneity in Finsler geometry and the positive curvature problem. Osaka J. Math. 55 (2018), no. 1, 177--194.

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