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On the Riemann-Finsler Geometry of Tangent Bundle of Lie Groups with Two-Dimensional Commutator Subgroup

Ali Hatami Shahi, Hamid Reza Salimi Moghaddam

Abstract

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the relationship between the sectional curvatures of $G$ and those of $TG$. Next, we define a Randers metric on $G$ from a left-invariant Riemannian metric and a left-invariant vector field, and lift it vertically and completely to $TG$. We investigate the conditions under which this Randers metric is of Berwald and Douglas type, respectively, and compute the flag curvatures in the Berwald case. In an addendum, we discuss geodesic vectors and bi-invariant Riemannian metrics on these Lie groups, highlighting the special unimodularity conditions. Finally, we provide explicit formulas for the Riemannian curvature tensor on the tangent bundle of such a Lie group.

On the Riemann-Finsler Geometry of Tangent Bundle of Lie Groups with Two-Dimensional Commutator Subgroup

Abstract

We begin by studying the Riemannian geometry of the tangent Lie group associated with a Lie group whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on . We establish the relationship between the sectional curvatures of and those of . Next, we define a Randers metric on from a left-invariant Riemannian metric and a left-invariant vector field, and lift it vertically and completely to . We investigate the conditions under which this Randers metric is of Berwald and Douglas type, respectively, and compute the flag curvatures in the Berwald case. In an addendum, we discuss geodesic vectors and bi-invariant Riemannian metrics on these Lie groups, highlighting the special unimodularity conditions. Finally, we provide explicit formulas for the Riemannian curvature tensor on the tangent bundle of such a Lie group.
Paper Structure (4 sections, 14 theorems, 109 equations, 1 table)

This paper contains 4 sections, 14 theorems, 109 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Riemann-Finsler Geometry of Tangent Lie Groups Having Two-Dimensional Commutator Subgroups
  4. Addendum

Key Result

Proposition 2.1

19. For an $n$-dimensional Lie group with a two-dimensional commutator subgroup equipped with a left-invariant metric $g$, the Levi-Civita connection and the sectional curvatures are given by: where $u,v\in \mathcal{P}$, and where $u,v\in \mathcal{P}$ are arbitrary orthonormal vectors and ${\left\|\cdot \right\|}^2=\left\langle \cdot , \cdot \right\rangle$.

Theorems & Definitions (30)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 20 more