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On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

Masoumeh Hosseini, Hamid Reza Salimi Moghaddam

Abstract

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous $(α,β)$-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any $(α,β)$-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of $3$-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.

On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

Abstract

Recently, it is shown that each regular homogeneous Finsler space admits at least one homogeneous geodesic through any point . The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous -spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any -homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of -dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.

Paper Structure

This paper contains 4 sections, 13 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous geodesic in homogeneous Kropina spaces
  4. Homogeneous Geodesics in Homogeneous $(\alpha,\beta)$-Spaces

Key Result

Proposition 2.1

Suppose that $(M,F)$ is a Kropina space which arises from a navigation data $(h,W)$. Then the isometry group of $(M,F)$ is a closed subgroup of the isometry group of Riemannian manifold $(M,h)$.

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • ...and 17 more