Paper · Aug 14, 2018

Some results on -Ricci Soliton and gradient -Einstein soliton in a complete Riemannian manifold

Absos Ali Shaikh, Chandan Kumar Mondal

math.DGarXiv:1808.04789v2
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Abstract

The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient -Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient -Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost -Ricci soliton (see Theorem 2).

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Submitted
Aug 14, 2018
Last revised
Aug 17, 2018 (v2)
Subjects
math.DG

Comments: 9 pages. We highly appreciate valuable comments from the interested researchers

Ricci solitonsAlmost & η-Ricci solitonsScalar curvature

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 On Ricci solitons whose potential is convex (Chandan Kumar Mondal and Absos Ali Shaikh, 2019)5Mondal 2019 Isometry theorem of Cartan-Hadamard manifold (Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al., 2020)6Shaikh 2020 Isometry theorem of gradient Shrinking Ricci solitons (Absos Ali Shaikh and Chandan Kumar Mondal, 2020)2Shaikh 2020 Compact gradient ρ-Einstein soliton is isometric to the Euclidean sphere (Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal, 2020)1Shaikh 2020 Some characterizations of ρ-Einstein solitons (Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal, 2020)3Shaikh 2020 Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton (Kiran Meena and Akhilesh Yadav, 2022)7Meena 2022 Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes (Absos Ali Shaikh and Biswa Ranjan Datta, 2022)8Shaikh 2022 Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds (Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya, 2023)4Ghosh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Compact gradient -Einstein soliton is isometric to the Euclidean sphere

    Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Chandan Kumar Mondal · Shared terms “Einstein soliton”, “Euclidean sphere”, “complete Riemannian”

  2. Isometry theorem of gradient Shrinking Ricci solitons

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Chandan Kumar Mondal · Shared terms “Euclidean sphere”

  3. Some characterizations of -Einstein solitons

    Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal

    Topics Scalar curvature, Ricci solitons · Same author Absos Ali Shaikh · Shared terms “Einstein soliton”, “Euclidean sphere”, “vector field”

  4. Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds

    Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “Einstein soliton”, “manifold admits”

  5. On Ricci solitons whose potential is convex

    Chandan Kumar Mondal and Absos Ali Shaikh

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Chandan Kumar Mondal

  6. Isometry theorem of Cartan-Hadamard manifold

    Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al.

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Chandan Kumar Mondal · Shared terms “manifolds satisfying”, “Riemannian manifold”, “Euclidean”

  7. Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton

    Kiran Meena and Akhilesh Yadav

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “manifold admits”, “Riemannian manifold”, “sufficient conditions”

  8. Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes

    Absos Ali Shaikh and Biswa Ranjan Datta

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Absos Ali Shaikh · Shared terms “possesses”

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) 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) 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)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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