What question did this study set out to answer?

This paper aims to establish conditions for Lie groups to support algebraic Ricci solitons and classify them in three dimensions.

March 29, 2026Open Access

Left Invariant Ricci Solitons on Three-Dimensional Lie Groups

Key Points

This paper aims to establish conditions for Lie groups to support algebraic Ricci solitons and classify them in three dimensions.
Deriving necessary and sufficient conditions for Lie groups to admit Ricci solitons.
Classifying algebraic Ricci solitons on three-dimensional real Lie groups.
Analyzing both unimodular and nonunimodular Lie groups for the presence of Ricci solitons.
Identified three-dimensional solvable and simple Lie groups lacking any algebraic Ricci soliton.
Discovered both unimodular and nonunimodular Lie groups that possess left invariant Ricci solitons.
Provided explicit solutions for the Ricci soliton equation of unimodular solvable Lie groups.

Abstract

We give a necessary and sufficient condition for an arbitrary real Lie group, to admit an algebraic Ricci soliton.As an application, we classify all algebraic Ricci solitons on threedimensional real Lie groups, up to automorphism.This classification shows that, in dimension three, there exist a solvable Lie group and a simple Lie group such that they do not admit any algebraic Ricci soliton.Also it is shown that there exist three-dimensional unimodular and nonunimodular Lie groups with left invariant Ricci solitons.Finally, for a unimodular solvable Lie group, the solution of the Ricci soliton equation is given, explicitly.

Bookmark

View Full Paper