Paper · Nov 1, 2026

Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

Vicente Cortés, Shubham Dwivedi, Lorenz Jetter

math.DGarXiv:2611.02062
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove that on a connected, simply-connected nilpotent Lie group, the notions of left-invariant generalized Ricci soliton, semi-algebraic and algebraic generalized Ricci soliton coincide. Moreover, every such soliton has harmonic torsion. We classify all algebraic solitons with positive definite metric on the five-dimensional Heisenberg algebra, the four-dimensional oscillator algebra and the four-dimensional split-oscillator algebra up to automorphisms and rescalings. We find that there is only a finite number of algebraic solitons, all of which have harmonic torsion.

View PDF View HTML arXiv page
Submitted
Nov 1, 2026
Subjects
math.DG
Generalized & G₂ flowsRicci 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 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)5Gindi 2021 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)7Fusi 2024 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)4Giudice 2025 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)6Ayad 2026 Four-dimensional Lorentzian algebraic Ricci solitons (Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo, 2026)8Garcia-Rio 2026 Harmonicity and Existence of Algebraic Generalized Ricci Solitons (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)3Chen 2026 Solvable Algebraic Generalized Ricci Solitons in low dimensions (Lorenzo Mucciante, 2026)2Mucciante 2026 Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)1Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “harmonic torsion”, “algebraic generalized”, “Lie”

  2. Solvable Algebraic Generalized Ricci Solitons in low dimensions

    Lorenzo Mucciante

    Topics Generalized & G₂ flows, 4-manifolds, Homogeneous spaces · Shared terms “algebraic generalized”, “four dimensional”, “Lie”

  3. Harmonicity and Existence of Algebraic Generalized Ricci Solitons

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “algebraic generalized”, “harmonic torsion”, “nilpotent Lie”

  4. Positive Hermitian curvature flow on 2-step nilpotent Lie groups

    Ettore Lo Giudice

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “algebraic soliton”, “semi algebraic”, “nilpotent Lie”

  5. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets

    Topics Generalized & G₂ flows, 4-manifolds, Homogeneous spaces · Shared terms “four dimensional”, “nilpotent”, “Lie”

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

    Youssef Ayad

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

  7. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Shared terms “harmonic torsion”, “invariant generalized”, “Lie groups”

  8. Four-dimensional Lorentzian algebraic Ricci solitons

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

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

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.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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.