Paper · May 11, 2017

On warped product gradient -Ricci solitons

Adara M. Blaga

math.DGarXiv:1705.04092v4
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Abstract

If the potential vector field of an -Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a Laplacian equation satisfied by the potential function . In a particular case of irrotational potential vector field we prove that the soliton is completely determined by . We give a way to construct a gradient -Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.

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Submitted
May 11, 2017
Last revised
Aug 23, 2020 (v4)
Subjects
math.DG
Journal
Filomat 31(18), 5791-5801 (2017)

Comments: 18 pages

Ricci solitonsAlmost & η-Ricci solitonsScalar curvatureRotational symmetry & warped products

Its neighbourhood

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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 Solitons and geometrical structures in a perfect fluid spacetime (Adara M. Blaga, 2017)7Blaga 2017 Almost η-Ricci solitons in (LCS)ₙ-manifolds (Adara M. Blaga, 2017)6Blaga 2017 Gradient Estimates on Warped Product Gradient Almost Ricci Solitons (Willian Isao Tokura, Levi Adriano, Romildo Pina et al., 2019)1Tokura 2019 Geometric solitons in a D-homothetically deformed Kenmotsu manifold (Adara M. Blaga, 2020)5Blaga 2020 Remarks on almost Riemann solitons with gradient or torse-forming vector field (Adara M. Blaga, 2020)3Blaga 2020 Characteristics of conformal Ricci soliton on warped product spaces (Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya, 2021)2Ganguly 2021 On conformal collineation and almost Ricci solitons (Adara M. Blaga and Bang-Yen Chen, 2022)8Blaga 2022 On some 3-dimensional almost η-Ricci solitons with diagonal metrics (Adara M. Blaga, 2024)4Blaga 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient Estimates on Warped Product Gradient Almost Ricci Solitons

    Willian Isao Tokura, Levi Adriano, Romildo Pina et al.

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “product gradient”, “Laplacian”

  2. Characteristics of conformal Ricci soliton on warped product spaces

    Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “potential vector”, “vector field”

  3. Remarks on almost Riemann solitons with gradient or torse-forming vector field

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Adara M. Blaga · Shared terms “vector field”, “Bochner formula”, “potential vector”

  4. On some -dimensional almost -Ricci solitons with diagonal metrics

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Adara M. Blaga · Shared terms “potential vector”, “vector field”

  5. Geometric solitons in a -homothetically deformed Kenmotsu manifold

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Same author Adara M. Blaga · Shared terms “vector field”, “potential vector”, “lower bound”

  6. Almost -Ricci solitons in -manifolds

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Adara M. Blaga · Shared terms “Bochner”

  7. Solitons and geometrical structures in a perfect fluid spacetime

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Adara M. Blaga · Shared terms “potential vector”, “gradient type”, “vector field”

  8. On conformal collineation and almost Ricci solitons

    Adara M. Blaga and Bang-Yen Chen

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Adara M. Blaga · Shared terms “potential vector”, “vector 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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.