Paper · Jul 28, 2026

Ricci Solitons on the Lie Group and Applications

Mohammed El Amine Mekki, Ahmed Mohammed Cherif

math.DGarXiv:2607.26240
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Abstract

In this paper, we study Ricci solitons on the three-dimensional solvable Lie group equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from into Euclidean spaces.

Ricci solitons3-manifoldsHomogeneous spacesHarmonic maps & DeTurck trick

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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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)6Bakhshandeh- 2021 Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (Carolyn Gordon and Michael Jablonski, 2021)5Gordon 2021 Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two (Hamid Reza Salimi Moghaddam, 2026)3Moghaddam 2026 On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry F⁴ (Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif, 2026)1Boukhari 2026 On the non-existence of symphonic maps (Ahmed Mohammed Cherif and Kaddour Zegga, 2026)8Cherif 2026 Solvable Lie algebras obtained by quivers and Einstein metrics (Fumika Mizoguchi, 2026)7Mizoguchi 2026 Ricci-Yamabe solitons on the Lie group Sol × ℝⁿ (Abdou Bousso and Ameth Ndiaye, 2026)2Bousso 2026 A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups (Fanqi Lin, 2026)4Lin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

    Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif PDF

    Topics Harmonic maps & DeTurck trick, Homogeneous spaces, Ricci solitons · Same author Ahmed Mohammed Cherif · Shared terms “Lie groups”, “invariant Riemannian”, “vector field”

  2. Ricci-Yamabe solitons on the Lie group

    Abdou Bousso and Ameth Ndiaye PDF

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

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

    Hamid Reza Salimi Moghaddam PDF

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

  4. A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups

    Fanqi Lin PDF

    Topics 3-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”, “left invariant”, “soliton vector”

  5. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski PDF

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

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

    Rohollah Bakhshandeh-Chamazkoti PDF

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

  7. Solvable Lie algebras obtained by quivers and Einstein metrics

    Fumika Mizoguchi PDF

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

  8. On the non-existence of symphonic maps

    Ahmed Mohammed Cherif and Kaddour Zegga PDF

    Topics Ricci solitons · Same author Ahmed Mohammed Cherif · Shared terms “vector field”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.