Paper · Sep 28, 2018

Some results on Ricci-Bourguignon solitons and almost solitons

Shubham Dwivedi

math.DGarXiv:1809.11103v2
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Abstract

We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon solitons and prove some results about them which generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci-Bourguignon solitons and compact gradient Ricci-Bourguignon almost solitons. Finally, using the integral formula we show that a compact gradient Ricci-Bourguignon almost soliton is isometric to an Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

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Submitted
Sep 28, 2018
Last revised
Sep 29, 2020 (v2)
Subjects
math.DG
Journal
Can. Math. Bull. 64 (2021) 591-604
DOI
10.4153/S0008439520000673

Comments: v2-added two examples of Ricci-Bourguignon almost solitons and some minor cosmetic changes, including a change in the title. 10 pages. Final version, to appear in Canadian Mathematical Bulletin

Ricci flowRicci solitonsAlmost & η-Ricci solitonsScalar curvature

Its neighbourhood

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  1. Geometric study of non-constant vector fields making hyperbolic space Ricci-Bourguignon solitons

    Mafal Ndiaye Diop, Abdou Bousso, Cheikh Khoule et al.

    Topics Almost & η-Ricci solitons · Shared terms “Ricci Bourguignon”, “Bourguignon soliton”, “vector field”

  2. Characterizations of Almost Ricci Bourguignon Solitons

    Mohammad Aqib, Hemangi Madhusudan Shah and Dhriti Sundar Patra

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “Ricci Bourguignon”, “Bourguignon soliton”

  3. Ricci Solitons on the Poincaré upper half plane

    Abdou Bousso and Ameth Ndiaye

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “Ricci Bourguignon”, “Bourguignon soliton”

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

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “Euclidean sphere”, “vector field”

  5. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “constant scalar”

  6. Ricci Almost Solitons on semi-Riemannian Warped Products

    Keti Tenenblat and Valter Borges

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “vector field”

  7. Clairaut conformal submersions from Ricci solitons

    Murat Polat

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “vector field”, “gradient Ricci”

  8. 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

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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.