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Riemannian Geometry of Lie Groups with One and Two-Dimensional Commutator Subgroups

Hamid Reza Salimi Moghaddam

TL;DR

The paper investigates left-invariant Riemannian metrics on Lie groups with one- and two-dimensional commutator subgroups, focusing on Ricci solitons. It provides explicit Levi-Civita connections, sectional and Ricci curvatures, and derives computable necessary-and-sufficient conditions for Ricci solitons in both the one- and two-dimensional commutator cases, including a complete classification in the one-dimensional case. For the two-dimensional case, it delivers a detailed algebraic criterion in terms of four endomorphisms and several derived quantities, and it shows that all indecomposable, five-or-more-dimensional examples in this setting fail to be Ricci solitons. Collectively, the results clarify when homogeneous Ricci solitons exist on solvable Lie groups with small derived subalgebras and provide obstructions for a broad class of indecomposable Lie groups.

Abstract

In this paper, we investigate left invariant Riemannian metrics on Lie groups with one and two-dimensional commutator subgroups. We explicitly provide the Levi-Civita connection, sectional curvature, and Ricci curvature, and we give computable necessary and sufficient conditions for these Riemannian manifolds to be Ricci solitons. Furthermore, we characterize all Ricci solitons on Lie groups with one-dimensional commutator subgroups. In the final section, we examine the results concerning all indecomposable Lie groups with two-dimensional commutator subgroups of dimension greater than or equal to five.

Riemannian Geometry of Lie Groups with One and Two-Dimensional Commutator Subgroups

TL;DR

The paper investigates left-invariant Riemannian metrics on Lie groups with one- and two-dimensional commutator subgroups, focusing on Ricci solitons. It provides explicit Levi-Civita connections, sectional and Ricci curvatures, and derives computable necessary-and-sufficient conditions for Ricci solitons in both the one- and two-dimensional commutator cases, including a complete classification in the one-dimensional case. For the two-dimensional case, it delivers a detailed algebraic criterion in terms of four endomorphisms and several derived quantities, and it shows that all indecomposable, five-or-more-dimensional examples in this setting fail to be Ricci solitons. Collectively, the results clarify when homogeneous Ricci solitons exist on solvable Lie groups with small derived subalgebras and provide obstructions for a broad class of indecomposable Lie groups.

Abstract

In this paper, we investigate left invariant Riemannian metrics on Lie groups with one and two-dimensional commutator subgroups. We explicitly provide the Levi-Civita connection, sectional curvature, and Ricci curvature, and we give computable necessary and sufficient conditions for these Riemannian manifolds to be Ricci solitons. Furthermore, we characterize all Ricci solitons on Lie groups with one-dimensional commutator subgroups. In the final section, we examine the results concerning all indecomposable Lie groups with two-dimensional commutator subgroups of dimension greater than or equal to five.
Paper Structure (4 sections, 5 theorems, 35 equations, 4 tables)

This paper contains 4 sections, 5 theorems, 35 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Left invariant Riemannian metrics on Lie groups with one-dimensional commutator subgroups
  3. Left invariant Riemannian metrics on Lie groups with two-dimensional commutator subgroups
  4. Indecomposable Lie groups

Key Result

Theorem 2.1

A Lie group $G\in\textsf{Lie(n,1)}$ equipped with an arbitrary Riemannian metric $\langle . , .\rangle$ is a Ricci soliton if and only if

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • ...and 1 more