Paper · Apr 15, 2016

Some geometrical properties of the Oscillator group

Yadollah Aryanejad

math.DGarXiv:1604.04369v2
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As on arXiv

Abstract

We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also determined, and the energy of critical points is explicitly calculated.

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Submitted
Apr 15, 2016
Last revised
Oct 18, 2019 (v2)
Subjects
math.DG
Journal
Caspian Journal of Mathematical Sciences, CJMS. 9 (2) (2020), 284-293
DOI
10.22080/CJMS.2020.17566.1441

Comments: arXiv admin note: substantial text overlap with arXiv:1412.6336, arXiv:1410.7626

Ricci solitonsHomogeneous spacesHarmonic maps & DeTurck trickPhysics

Its neighbourhood

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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 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 On the stability of harmonic maps under the homogeneous Ricci flow (Rafaela F. do Prado and Lino Grama, 2017)1Prado 2017 On the nonexistence of harmonic and bi-harmonic maps (Ahmed Mohammed Cherif, 2020)4Cherif 2020 A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups (Yuji Kondo and Hiroshi Tamaru, 2020)5Kondo 2020 Perelman's Ricci Flow in Topological Quantum Gravity (Alexander Frenkel, Petr Horava and Stephen Randall, 2020)6Frenkel 2020 Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)8Bakhshandeh- 2021 On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry F⁴ (Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif, 2026)3Boukhari 2026 Ricci Solitons on the Lie Group Sol³_(m,n) and Applications (Mohammed El Amine Mekki and Ahmed Mohammed Cherif, 2026)2Mekki 2026 Generalized Ricci Solitons on Heisenberg and Oscillator Algebras (Vicente Cortés, Shubham Dwivedi and Lorenz Jetter, 2026)7Cortés 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Harmonic maps & DeTurck trick, Homogeneous spaces

  2. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki and Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Homogeneous spaces, Ricci solitons · Shared terms “vector field”, “equipped”

  3. On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

    Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Homogeneous spaces, Ricci solitons · Shared terms “vector field”

  4. On the nonexistence of harmonic and bi-harmonic maps

    Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Ricci solitons · Shared terms “vector field”

  5. A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups

    Yuji Kondo and Hiroshi Tamaru

    Topics Physics, Homogeneous spaces, Ricci solitons · Shared terms “invariant Lorentzian”

  6. Perelman's Ricci Flow in Topological Quantum Gravity

    Alexander Frenkel, Petr Horava and Stephen Randall

    Topics Harmonic maps & DeTurck trick, Physics, Ricci solitons

  7. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

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

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

    Rohollah Bakhshandeh-Chamazkoti

    Topics Physics, Homogeneous spaces, Ricci solitons · Shared terms “invariant Lorentzian”

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) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.