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$\delta$-homogeneity in Finsler geometry and the positive curvature problem
Differential Geometry · Submitted 3 Nov 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 $G$-$\delta$-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).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we explore the similarity between normal homogeneity and $\delta$-homogeneity in Finsler geometry. | contribution 0.77 | 0.82 | 0.05 | asserted 0.60 |
| s1 | They are both non-negatively curved Finsler spaces. | background 0.82 | 0.16 | 0.04 | asserted 1.00 |
| s2 | We show that any connected $\delta$-homogeneous Finsler space is $G$-$\delta$-homo-geneous, for some suitably chosen connected quasi-compact $G$. | result 0.98 | 0.94 | 0.20 | asserted 1.00 |
| s3 | 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. | result 0.40 | 0.77 | 0.13 | asserted 0.98 |
| s4 | 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}). | result 0.95 | 0.93 | 0.63 | asserted 1.00 |
| s5 | 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. | result 0.80 | 0.87 | 0.41 | asserted 0.98 |
| s6 | As a by-product, this argument tells more about $\delta$-homogeneous Finsler metrics satisfying the (FP) condition (a weaker version of positively curved condition). | interpretation 0.50 | 0.88 | 0.06 | asserted 0.93 |
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Original 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).
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How the literature uses it
2 arXiv papers cite this in 4 sentences
What it cites · 16 sentences without a confident label
Uses the method or tool of 1 paper
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In this paper, we will use Theorem [ref], and the method in [ref] to study the classification of positively curved $δ$-homogeneous Finsler spaces.
Normal homogeneous Finsler spaces · Introduction · 0.95
Cites as background 1 paper
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Then according to [ref] , a subalgebra $𝔰⊂𝔤$ is called a flat splitting subalgebra (FSS in short), when it satisfies the following conditions: * (1) $𝔰$ is the intersection of a family of Cartan subalgebras of $𝔤$. * (2) $𝔰=𝔰∩𝔥+ 𝔰∩𝔪$. * (3) $𝔰∩𝔪≥2$.
Normal homogeneous Finsler spaces · The positive curvature problem for $δ$-homogeneous Finsler spaces · 0.80
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