Paper · Mar 11, 2020

Compact gradient -Einstein soliton is isometric to the Euclidean sphere

Absos Ali Shaikh, Chandan Kumar Mondal, Prosenjit Mandal

math.DGarXiv:2003.05234
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Abstract

In this paper we have investigated some aspects of gradient -Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient -Einstein soliton is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have showed that in a non-compact gradient -Einstein soliton satisfying some integral condition, the scalar curvature vanishes.

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Submitted
Mar 11, 2020
Subjects
math.DG

Comments: 6 pages

Ricci solitonsScalar curvatureBoundaries & non-compact

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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 Some results on η-Ricci Soliton and gradient ρ-Einstein soliton in a complete Riemannian manifold (Absos Ali Shaikh and Chandan Kumar Mondal, 2018)1Shaikh 2018 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)3Shaikh 2020 Some characterizations of ρ-Einstein solitons (Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal, 2020)2Shaikh 2020 Rigidity results for shrinking and expanding Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)8Leandro 2022 Triviality Results and Conjugate Radius Estimation of Ricci Solitons (Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu, 2023)4Shaikh 2023 Conjugate radius and parabolicity along the Ricci flow (Chandan Kumar Mondal, 2026)7Mondal 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some results on -Ricci Soliton and gradient -Einstein soliton in a complete Riemannian manifold

    Absos Ali Shaikh and Chandan Kumar Mondal

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

  2. Some characterizations of -Einstein solitons

    Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Prosenjit Mandal · Shared terms “Einstein soliton”, “Euclidean sphere”

  3. 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”, “soliton satisfying”

  4. Triviality Results and Conjugate Radius Estimation of Ricci Solitons

    Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu

    Topics Boundaries & non-compact, Ricci solitons · Same authors Absos Ali Shaikh, Prosenjit Mandal · Shared terms “compact gradient”

  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 · Shared terms “integral condition”

  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, Prosenjit Mandal · Shared terms “integral condition”, “soliton satisfying”, “Euclidean”

  7. Conjugate radius and parabolicity along the Ricci flow

    Chandan Kumar Mondal

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Same author Chandan Kumar Mondal

  8. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Shared terms “compact gradient”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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    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.