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Two-step homogeneous geodesics in some homogeneous Finsler manifolds

Masoumeh Hosseini, Hamid Reza Salimi Moghaddam

Abstract

A natural extension of a homogeneous geodesic in homogeneous Riemannian spaces $G/H$, known as a two-step homogeneous geodesic, can be expressed of the form $γ(t)=π(\exp(tx)\exp(ty))$, where $x$ and $y$ are elements of the Lie algebra of $G$. This paper aims to expand this concept to homogeneous Finsler spaces. We provide certain sufficient conditions for $(α,β)$ spaces and decomposable cubic spaces to possess a one-parameter family of invariant Finsler metrics that can be classified as two-step Finsler geodesic orbit spaces. Additionally, we present some illustrative examples of these spaces.

Two-step homogeneous geodesics in some homogeneous Finsler manifolds

Abstract

A natural extension of a homogeneous geodesic in homogeneous Riemannian spaces , known as a two-step homogeneous geodesic, can be expressed of the form , where and are elements of the Lie algebra of . This paper aims to expand this concept to homogeneous Finsler spaces. We provide certain sufficient conditions for spaces and decomposable cubic spaces to possess a one-parameter family of invariant Finsler metrics that can be classified as two-step Finsler geodesic orbit spaces. Additionally, we present some illustrative examples of these spaces.

Paper Structure

This paper contains 4 sections, 10 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Naturally reductive Finsler spaces and geodesic vectors
  4. Two-step homogeneous g.o. Finsler spaces

Key Result

Proposition 3.1

Let $(M=G/H, F)$ be a homogeneous Finsler space with a reductive decomposition $\mathfrak{g}=\mathfrak{h} \oplus \mathfrak{m}$, where $F$ is an invariant $(\alpha,\beta)$-metric characterized by an invariant Riemannian metric $h$ and an invariant vector field $X$. Assume that $0\neq y \in \mathfrak{

Theorems & Definitions (26)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 4.1
  • Definition 4.2
  • ...and 16 more