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

TL;DR

This paper explores the geometry of Lie groups with small commutator subgroups, providing explicit formulas for curvature and conditions for Ricci solitons, including classifications for specific cases.

Contribution

It offers explicit computations of Levi-Civita connection, curvature, and Ricci solitons on Lie groups with low-dimensional commutator subgroups, and characterizes all Ricci solitons in these cases.

Findings

Explicit formulas for Levi-Civita connection, sectional, and Ricci curvature.

Necessary and sufficient conditions for Ricci solitons.

Complete classification of Ricci solitons on certain 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.