Paper · Nov 18, 2020

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

Yuji Kondo, Hiroshi Tamaru

math.DGarXiv:2011.09118
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Abstract

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension with , and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.

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Submitted
Nov 18, 2020
Subjects
math.DG

Comments: 32 pages

Ricci solitons3-manifoldsHomogeneous spacesPhysics

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

    Rohollah Bakhshandeh-Chamazkoti

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

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    Topics Physics, 3-manifolds, Ricci solitons · Shared terms “invariant Lorentzian”, “Lorentzian metrics”, “Heisenberg group”

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

    Mohamed Boucetta and Abdelmounaim Chakkar

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

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

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

  5. Nilpotent Lie algebras obtained by quivers and Ricci solitons

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    Topics Homogeneous spaces, Ricci solitons · Same author Hiroshi Tamaru · Shared terms “nilpotent Lie”, “left invariant”, “Lie groups”

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

    Yong Wang

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

  7. 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”

  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) 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) 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)

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