Paper · Sep 24, 2022

The extended quasi-Einstein manifolds with generalised Ricci solitons

Zhiming Huang, Weijun Lu, Fuhong Su

math.DGarXiv:2209.11921
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Abstract

As a generalization of Einstein manifolds, the nearly quasi-Einstein manifolds and pseudo quasi-Einstein manifolds are both interesting and useful in studying the general relativity. In this paper, we study the extended quasi-Einstein manifolds which derive from pseudo quasi-Einstein manifolds. After showing the existence theorem of extended quasi-Einstein manifold, we give some special geometric properties of such manifolds. At the same time, we also discuss the extended quasi-Einstein manifolds with certain soliton like generalised Ricci soliton or Riemann soliton. Furthermore, we construct some nontrivial example to illustrate these extended quasi-Einstein manifolds.

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Submitted
Sep 24, 2022
Subjects
math.DG

Comments: 16 pages

Generalized & G₂ flowsRicci solitonsEinstein metricsAlmost & η-Ricci solitonsPhysics

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  1. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “generalization”

  2. On curvature related geometric properties of Hayward black hole spacetime

    Absos Ali Shaikh, Shyamal Kumar Hui, Biswa Ranjan Datta et al.

    Topics Almost & η-Ricci solitons, Einstein metrics, Physics · Shared terms “quasi Einstein”, “geometric properties”

  3. Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds

    Samuel Belo

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  4. Symmetry and pseudosymmetry properties of Vaidya-Bonner-de Sitter spacetime

    Absos Ali Shaikh, Shyamal kumar Hui, Mousumi Sarkar et al.

    Topics Almost & η-Ricci solitons, Einstein metrics, Physics · Shared terms “quasi Einstein”, “interesting”, “generalization”

  5. Direct proofs for spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  6. On the Generalised Ricci Solitons and Sasakian Manifolds

    Ahmed Mohammed Cherif, Kaddour Zegga and Gherici Beldjilali

    Topics Generalized & G₂ flows, Ricci solitons · Shared terms “generalised Ricci”

  7. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  8. Characterizations of Perfect fluid spacetimes obeying -gravity equipped with different gradient solitons

    Krishnendu De Young Jin Suh and Uday Chand De

    Topics Almost & η-Ricci solitons, Physics, Ricci solitons · Shared terms “quasi Einstein”

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.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 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.