Paper · Feb 25, 2021

The classification of left-invariant para-Kähler structures on simply connected four-dimensional Lie groups

Wadia Mansouri, Ahmad Oufkou

math.SGarXiv:2102.12710v3
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As on arXiv

Abstract

We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.

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Submitted
Feb 25, 2021
Last revised
Apr 19, 2021 (v3)
Subjects
math.SG, math.DG
Ricci solitonsRicci-flat & Calabi–Yau4-manifoldsKähler geometryHomogeneous spaces

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    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”

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    Topics 4-manifolds, Homogeneous spaces, Kähler geometry · Shared terms “connected four”, “simply connected”, “four dimensional”

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

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

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    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces, Ricci solitons · Shared terms “Lie groups”, “left invariant”, “simply connected”

  7. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

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

  8. Parallel spinor flows on three-dimensional Cauchy hypersurfaces

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    Topics Ricci-flat & Calabi–Yau, 4-manifolds, Homogeneous spaces · Shared terms “left invariant”, “connected Lie”, “simply connected”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.