Paper · Mar 12, 2019

Lorentzian left invariant metrics on three dimensional unimodular Lie groups and their curvatures

Mohamed Boucetta, Abdelmounaim Chakkar

math.DGarXiv:1903.05194
View PDF View HTML arXiv page

As on arXiv

Abstract

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group , the special unitary group , the universal covering group of the special linear group, the solvable Lie group and the universal covering group of the connected component of the Euclidean group. For each among these Lie groups, we give explicitly the list of all Lorentzian left invariant metrics on , up to un automorphism of . Moreover, for any Lorentzian left invariant metric in this list we give its Ricci curvature, scalar curvature, the signature of the Ricci curvature and we exhibit some special features of these curvatures. Namely, we give all the metrics with constant curvature, semi-symmetric non locally symmetric metrics and the Ricci solitons.

View PDF View HTML arXiv page
Submitted
Mar 12, 2019
Subjects
math.DG

Comments: 10 pages submitted

Ricci solitons3-manifoldsScalar curvatureHomogeneous spacesPhysics

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 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)4Lafuente 2018 Canonical connections and algebraic Ricci solitons of three-dimensional Lorentzian Lie groups (Yong Wang, 2020)8Wang 2020 A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups (Yuji Kondo and Hiroshi Tamaru, 2020)1Kondo 2020 Affine Ricci solitons of three-dimensional Lorentzian Lie groups (Yong Wang, 2020)7Wang 2020 Affine Ricci Solitons associated to the Bott connection on three-dimensional Lorentzian Lie groups (Tong Wu and Yong Wang, 2021)6Wu 2021 Classification of Left Invariant Riemannian metrics on Complex hyperbolic space (Andrijana Dekic, Marijana Babic and Srdjan Vukmirovic, 2021)5Dekic 2021 Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)2Bakhshandeh- 2021 A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups (Fanqi Lin, 2026)3Lin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups

    Yuji Kondo and Hiroshi Tamaru

    Topics Physics, 3-manifolds, Homogeneous spaces · Shared terms “left invariant”, “Lie groups”, “Lorentzian”

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

    Rohollah Bakhshandeh-Chamazkoti

    Topics Physics, Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “nilpotent Lie”

  3. A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups

    Fanqi Lin

    Topics Physics, 3-manifolds, Homogeneous spaces · Shared terms “left invariant”, “Lie groups”, “Lorentzian”

  4. Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

    Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni

    Topics Homogeneous spaces · Shared terms “unimodular Lie”, “invariant metrics”, “Lie groups”

  5. Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

    Andrijana Dekic, Marijana Babic and Srdjan Vukmirovic

    Topics Homogeneous spaces, Scalar curvature, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “invariant metric”

  6. Affine Ricci Solitons associated to the Bott connection on three-dimensional Lorentzian Lie groups

    Tong Wu and Yong Wang

    Topics Physics, 3-manifolds, Homogeneous spaces · Shared terms “Lie groups”, “Lorentzian”

  7. Affine Ricci solitons of three-dimensional Lorentzian Lie groups

    Yong Wang

    Topics Physics, 3-manifolds, Homogeneous spaces · Shared terms “Lie groups”, “Lorentzian”

  8. Canonical connections and algebraic Ricci solitons of three-dimensional Lorentzian Lie groups

    Yong Wang

    Topics Physics, 3-manifolds, Homogeneous spaces · Shared terms “Lie groups”, “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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 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.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.