Paper · Jan 5, 2026

Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

Hamid Reza Salimi Moghaddam

math.DGarXiv:2601.10730v2
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Abstract

In this article, we investigate when a left-invariant Riemannian metric on a Lie group is a Ricci soliton, under the assumption that the derived algebra has dimension at most two. We establish computable necessary and sufficient conditions for a given left-invariant Riemannian metric to be a Ricci soliton. As applications, we obtain several examples of Ricci nilsolitons and apply our results to indecomposable Lie groups of dimension at least five with two-dimensional derived algebra.

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Submitted
Jan 5, 2026
Last revised
Sep 12, 2026 (v2)
Subjects
math.DG
Ricci solitonsSurfacesHomogeneous spaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Hilbert–Mumford criterion for nilsolitons

    Yoshinori Hashimoto

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”, “nilsolitons”, “left invariant”

  2. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki and Ahmed Mohammed Cherif

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”, “left invariant”, “invariant Riemannian”

  3. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski

    Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “invariant Riemannian”

  4. Ricci-Yamabe solitons on the Lie group

    Abdou Bousso and Ameth Ndiaye

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”, “left invariant”, “invariant Riemannian”

  5. Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

    Rohollah Bakhshandeh-Chamazkoti

    Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”

  6. The mean curvature flow of subgroups on Lie groups of dimension three

    Romina M. Arroyo, Gabriela P. Ovando and Mariel SáezWider flows

    Topics Surfaces, Homogeneous spaces · Shared terms “Lie groups”, “left invariant”

  7. Nilpotent Lie algebras obtained by quivers and Ricci solitons

    Fumika Mizoguchi and Hiroshi Tamaru

    Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “invariant metrics”

  8. Nilpotent Lie algebras obtained by ordered sets and Ricci solitons

    Yihao Zheng and Shenglin Zhu

    Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “invariant metrics”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.