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On Naturally Reductive $\boldsymbol{(α_1,α_2)}$-Metrics

Ali Hatami Shahi, Hamid Reza Salimi Moghaddam

Abstract

In this paper, we investigate the converse of the Tan-Xu theorem, which states that the naturally reductive property of a Riemannian metric is inherited by a naturally reductive $(α_1,α_2)$-metric, and we show that, under certain conditions, the converse also holds. We also examine the relationship between geodesic vector fields on homogeneous Riemannian spaces and homogeneous $(α_1,α_2)$-spaces. Finally, we construct left-invariant $(α_1,α_2)$-metrics on the tangent bundle of Lie groups using left-invariant Randers metrics on the base Lie group, and study their geometric relations.

On Naturally Reductive $\boldsymbol{(α_1,α_2)}$-Metrics

Abstract

In this paper, we investigate the converse of the Tan-Xu theorem, which states that the naturally reductive property of a Riemannian metric is inherited by a naturally reductive -metric, and we show that, under certain conditions, the converse also holds. We also examine the relationship between geodesic vector fields on homogeneous Riemannian spaces and homogeneous -spaces. Finally, we construct left-invariant -metrics on the tangent bundle of Lie groups using left-invariant Randers metrics on the base Lie group, and study their geometric relations.
Paper Structure (3 sections, 7 theorems, 28 equations)

This paper contains 3 sections, 7 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the Natural Reductivity of $\boldsymbol{(\alpha_1,\alpha_2)}$-Metrics

Key Result

Theorem 2.4

11 Assume $\phi$ and $\psi$ are two positive functions on the interval $[0,1]$ such that $\phi (s) = \psi \left(\sqrt{1-s^2}\right)$. Also, let $F=\alpha \phi (\frac{\alpha _2}{\alpha} )=\alpha \psi (\frac{\alpha _1}{\alpha} )$ be a $(\alpha _1,\alpha _2)$-norm on $\mathbb{R}^n$ with dimension decom Conversely, if $\phi (s)-s\phi '(s)+(1- s^2)\phi "(s)>0$, then $F(\alpha )=\alpha \phi (\frac{\alph

Theorems & Definitions (23)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 3.1
  • ...and 13 more