Paper · Dec 19, 2019

On the classification of five-dimensional nilsolitons

Hamid Reza Salimi Moghaddam

math.DGarXiv:1912.13322v3
View PDF View HTML arXiv page

As on arXiv

Abstract

In 2002, using a variational method, Lauret classified five-dimensional nilsolitons. In this work, using the algebraic Ricci soliton equation, we obtain the same classification. We show that, among ten classes of five-dimensional nilmanifolds, seven classes admit Ricci soliton structure. In any case, the derivation which satisfies the algebraic Ricci soliton equation is computed.

View PDF View HTML arXiv page
Submitted
Dec 19, 2019
Last revised
May 20, 2021 (v3)
Subjects
math.DG
Journal
Journal of the Iranian Mathematical Society, Year: March 2024 Volume: 5 Issue: Pages: 243 - 252
DOI
10.30504/jims.2024.463565.1189
Ricci solitonsHomogeneous 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 2016, 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 Randers Ricci soliton homogeneous nilmanifolds (Hamid Reza Salimi Moghaddam, 2017)2Moghaddam 2017 A Hilbert–Mumford criterion for nilsolitons (Yoshinori Hashimoto, 2023)3Hashimoto 2023 Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two (Hamid Reza Salimi Moghaddam, 2026)1Moghaddam 2026 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)4Ayad 2026 Four-dimensional Lorentzian algebraic Ricci solitons (Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo, 2026)5Garcia-Rio 2026 Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups (Youssef Ayad, 2026)8Ayad 2026 Harmonicity and Existence of Algebraic Generalized Ricci Solitons (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)6Chen 2026 Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)7Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

    Hamid Reza Salimi Moghaddam

    Topics Homogeneous spaces, Ricci solitons · Same author Hamid Reza Salimi Moghaddam · Shared terms “nilsolitons”

  2. Randers Ricci soliton homogeneous nilmanifolds

    Hamid Reza Salimi Moghaddam

    Topics Homogeneous spaces, Ricci solitons · Same author Hamid Reza Salimi Moghaddam · Shared terms “nilmanifolds”

  3. A Hilbert–Mumford criterion for nilsolitons

    Yoshinori Hashimoto

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “algebraic”

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

    Youssef Ayad

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

  5. Four-dimensional Lorentzian algebraic Ricci solitons

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

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

  6. Harmonicity and Existence of Algebraic Generalized Ricci Solitons

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “algebraic”, “derivation”

  7. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “algebraic”

  8. Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

    Youssef Ayad

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

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) 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.