Paper · Jan 21, 2026

Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

Youssef Ayad

math.DGarXiv:2601.15008
PDF Abstract page

As on arXiv

Abstract

We investigate the conditions under which pseudo-Riemannian inner products induce pseudo-Riemannian algebraic Ricci solitons on four-dimensional Lie algebras. By analyzing the algebraic Ricci soliton equation for each four-dimensional Lie algebra, we obtain a complete description of when such pseudo-Riemannian algebraic Ricci solitons arise in dimension four. We present two applications of our formalism on a chosen four-dimensional Lie algebra by exhibiting a pseudo-Riemannian algebraic Ricci soliton and a flat pseudo-Riemannian inner product, which is a trivial algebraic Ricci soliton.

Ricci solitons4-manifoldsHomogeneous 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.

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 Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds (Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso, 2025)8Conti 2025 Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group (Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim, 2025)5Calvaruso 2025 Four-dimensional Lorentzian algebraic Ricci solitons (Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo, 2026)2Garcia-Rio 2026 Nilpotent Lie algebras obtained by ordered sets and Ricci solitons (Yihao Zheng and Shenglin Zhu, 2026)7Zheng 2026 Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups (Youssef Ayad, 2026)1Ayad 2026 Lorentzian Algebraic Ricci Solitons on the Heisenberg Group (Youssef Ayad, 2026)3Ayad 2026 Solvable Algebraic Generalized Ricci Solitons in low dimensions (Lorenzo Mucciante, 2026)4Mucciante 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

    Youssef Ayad PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Same author Youssef Ayad · Shared terms “algebraic Ricci”, “Lie groups”, “four dimensional”

  2. Four-dimensional Lorentzian algebraic Ricci solitons

    Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “algebraic Ricci”, “dimensional Lie”, “four dimensional”

  3. Lorentzian Algebraic Ricci Solitons on the Heisenberg Group

    Youssef Ayad PDF

    Topics Ricci solitons · Same author Youssef Ayad · Shared terms “algebraic Ricci”, “Lie algebras”, “inner product”

  4. Solvable Algebraic Generalized Ricci Solitons in low dimensions

    Lorenzo Mucciante PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “dimensional Lie”, “Lie algebras”, “algebraic”

  5. Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “four dimensional”, “Lie groups”, “description”

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

    Rohollah Bakhshandeh-Chamazkoti PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”

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

    Yihao Zheng and Shenglin Zhu PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “algebraic Ricci”, “Lie algebras”, “Lie groups”

  8. Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

    Diego Conti, Federico A. Rossi and Romeo Segnan Dalmasso PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “pseudo Riemannian”, “Lie algebras”, “Lie groups”

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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 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)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.