# The Existence of Two Homogeneous Geodesics in Finsler Geometry

## Abstract

**:**

## 1. Introduction

## 2. Basic Settings

**Lemma**

**1**

**.**Let $(G/H,F)$ be a homogeneous Finsler space with a reductive decomposition $\mathfrak{g}=\mathfrak{m}+\mathfrak{h}$. A nonzero vector $y\in \mathfrak{g}$ is geodesic if and only if it holds

## 3. The Main Result

**Theorem**

**1.**

**Proof.**

## Funding

## Conflicts of Interest

## References

- Dušek, Z. Homogeneous geodesics and g.o. manifolds. Note Mat.
**2018**, 38, 1–15. [Google Scholar] - Kowalski, O.; Szenthe, J. On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicata
**2000**, 81, 209–214, Erratum: Geom. Dedicata**2001**, 84, 331–332. [Google Scholar] [CrossRef] - Kowalski, O.; Nikčević, S.; Vlášek, Z. Homogeneous Geodesics in Homogeneous Riemannian Manifolds—Examples; Preprint Reihe Mathematik, TU Berlin, No. 665/2000; TU Berlin: Berlin, Germany, 2000. [Google Scholar]
- Kowalski, O.; Vlášek, Z. Homogeneous Riemannian manifolds with only one homogeneous geodesic. Publ. Math. Debrecen
**2003**, 62, 437–446. [Google Scholar] - Dušek, Z. The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds. J. Geom. Phys.
**2010**, 60, 687–689. [Google Scholar] [CrossRef] - Dušek, Z. The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds. Math. Nachr.
**2015**, 288, 872–876. [Google Scholar] [CrossRef] [Green Version] - Yan, Z.; Deng, S. Existence of homogeneous geodesics on homogeneous Randers spaces. Houston J. Math.
**2018**, 44, 481–493. [Google Scholar] - Dušek, Z. The affine approach to homogeneous geodesics in homogeneous Finsler spaces. Archivum Mathematicum (Brno)
**2018**, 54, 127–133. [Google Scholar] [CrossRef] - Dušek, Z. The existence of homogeneous geodesics in special homogeneous Finsler spaces. Matematički Vesnik
**2019**, 71, 16–22. [Google Scholar] - Yan, Z.; Huang, L. On the existence of homogeneous geodesic in homogeneous Finsler spaces. J. Geom. Phys.
**2018**, 124, 264–267. [Google Scholar] [CrossRef] - Dušek, Z. Homogeneous Randers spaces admitting just two homogeneous geodesics. Archivum Mathematicum (Brno)
**2019**, in press. [Google Scholar] - Shen, Z. Lectures on Finsler Geometry; World Scientific: Singapore, 2001. [Google Scholar]
- Bao, D.; Chern, S.-S.; Shen, Z. An Introduction to Riemann-Finsler Geometry; Springer Science+Business Media: New York, NY, USA, 2000. [Google Scholar]
- Deng, S. Homogeneous Finsler Spaces; Springer Science+Business Media: New York, NY, USA, 2012. [Google Scholar]
- Kowalski, O.; Vanhecke, L. Riemannian manifolds with homogeneous geodesics. Boll. Un. Math. Ital.
**1991**, 5, 189–246. [Google Scholar] - Latifi, D. Homogeneous geodesics in homogeneous Finsler spaces. J. Geom. Phys
**2007**, 57, 1421–1433. [Google Scholar] [CrossRef] [Green Version]

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dušek, Z.
The Existence of Two Homogeneous Geodesics in Finsler Geometry. *Symmetry* **2019**, *11*, 850.
https://doi.org/10.3390/sym11070850

**AMA Style**

Dušek Z.
The Existence of Two Homogeneous Geodesics in Finsler Geometry. *Symmetry*. 2019; 11(7):850.
https://doi.org/10.3390/sym11070850

**Chicago/Turabian Style**

Dušek, Zdeněk.
2019. "The Existence of Two Homogeneous Geodesics in Finsler Geometry" *Symmetry* 11, no. 7: 850.
https://doi.org/10.3390/sym11070850