Paper · Mar 8, 2021

Ricci solitons on contact pseudo-metric manifolds

Eftekhar Asgharzadeh, Morteza Faghfouri

math.DGarXiv:2103.05052
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Abstract

In this paper, we prove that a Sasakian pseudo-metric manifold which admits an Ricci soliton is an Einstein manifold, and if the potential vector field of the Ricci soliton is not a Killing vector field then the manifold is homothetically fixed, and the vector field leaves the structure tensor field invariant. Next, we prove that a contact pseudo-metric manifold with a gradient Ricci soliton metric is Einstein. Moreover, we study contact pseudo-metric manifolds admitting an Ricci soliton with a potential vector field point-wise colinear with the Reeb vector field. Finally, we study gradient Ricci solitons on -contact pseudo-metric manifolds.

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Submitted
Mar 8, 2021
Subjects
math.DG

Comments: 14 pages

Ricci solitonsEinstein metricsSasakian & contact geometry

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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 η-Ricci solitons in (ε)-almost paracontact metric manifolds (Adara Monica Blaga, Selcen Yüksel Perktaş, Bilal Eftal Acet et al., 2017)8Blaga 2017 The k-almost Ricci solitons and contact geometry (Amalendu Ghosh and Dhriti Sundar Patra, 2018)4Ghosh 2018 Certain results on Kenmotsu pseudo-metric manifolds (Devaraja Mallesha Naik, Venkatesha and D. G. Prakasha, 2018)1Naik 2018 Almost η-Ricci solitons on Kenmotsu manifolds (Dhriti Sundar Patra and Vladimir Rovenski, 2020)6Patra 2020 Remarks on Riemann and Ricci solitons in (α,β)-contact metric manifolds (Adara M. Blaga and Dan Radu Latcu, 2020)5Blaga 2020 On non-gradient (m,ρ)-quasi-Einstein contact metric manifolds (Dhriti Sundar Patra and Vladimir Rovenski, 2020)3Patra 2020 Geometry of almost contact metrics as almost *-Ricci solitons (Dhriti Sundar Patra, Akram Ali and Fatemah Mofarreh, 2021)2Patra 2021 Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds (Vladimir Rovenski, 2023)7Rovenski 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Certain results on Kenmotsu pseudo-metric manifolds

    Devaraja Mallesha Naik, Venkatesha and D. G. Prakasha

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “pseudo metric”, “manifold admitting”, “Next”

  2. Geometry of almost contact metrics as almost -Ricci solitons

    Dhriti Sundar Patra, Akram Ali and Fatemah Mofarreh

    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Shared terms “potential vector”, “vector field”, “Reeb vector”

  3. On non-gradient -quasi-Einstein contact metric manifolds

    Dhriti Sundar Patra and Vladimir Rovenski

    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Shared terms “vector field”, “potential vector”, “Reeb vector”

  4. The k-almost Ricci solitons and contact geometry

    Amalendu Ghosh and Dhriti Sundar Patra

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “vector field”, “point wise”, “potential vector”

  5. Remarks on Riemann and Ricci solitons in -contact metric manifolds

    Adara M. Blaga and Dan Radu Latcu

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “vector field”, “potential vector”

  6. Almost -Ricci solitons on Kenmotsu manifolds

    Dhriti Sundar Patra and Vladimir Rovenski

    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Shared terms “vector field”, “Reeb vector”, “potential vector”

  7. Generalized Ricci solitons and Einstein metrics on weak -contact manifolds

    Vladimir Rovenski

    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Shared terms “Killing vector”, “vector field”

  8. -Ricci solitons in -almost paracontact metric manifolds

    Adara Monica Blaga, Selcen Yüksel Perktaş, Bilal Eftal Acet et al.

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “potential vector”, “vector field”, “tensor field”

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) 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) Sasakian & contact geometry Sasakian manifolds are odd-dimensional manifolds whose metric cone is Kähler. They carry a transverse Kähler structure, so Kähler–Ricci flow has an… Start with: THE SASAKI–RICCI FLOW (SMOCZYK 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.