Paper · Mar 9, 2026

On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

Halima Boukhari, Hadjer Okbani, Ahmed Mohammed Cherif

math.DGarXiv:2603.08969
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Abstract

In this paper, we consider a left-invariant Riemannian metric on the Lie group . We classify Ricci solitons on and show that all such solitons are expanding and non-gradient. Moreover, we study the existence of harmonic maps from compact Riemannian manifolds into . Finally, we characterize a class of harmonic vector fields on .

Ricci solitonsGeometrizationHomogeneous 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.

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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. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki 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. 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”, “compact Riemannian”, “Riemannian manifolds”

  3. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec PDF

    Topics Geometrization, Homogeneous spaces · Shared terms “Thurston geometries”

  4. Ricci-Yamabe solitons on the Lie group

    Abdou Bousso and Ameth Ndiaye PDF

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

  5. Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes

    Tanay Gupta, Sukanta Panda and Rajib Saha PDF

    Topics Geometrization · Shared terms “Thurston geometries”

  6. 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 “left invariant”, “invariant Riemannian”, “Lie groups”

  7. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos PDF

    Topics Geometrization · Shared terms “Thurston”

  8. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al. PDF

    Topics Geometrization · Shared terms “Thurston”

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) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 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.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.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.