Paper · Mar 1, 2022

Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton

Kiran Meena, Akhilesh Yadav

math.DGarXiv:2203.00683v7Mediterranean Journal of Mathematics, Vol. 20, No. 3, Article 168 (2023)
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Abstract

In this paper, we study conformal submersions from Ricci solitons to Riemannian manifolds with non-trivial examples. First, we study some properties of the O'Neill tensor in the case of conformal submersion. We also find a necessary and sufficient condition for conformal submersion to be totally geodesic and calculate the Ricci tensor for the total manifold of such a map with different assumptions. Further, we consider a conformal submersion from a Ricci soliton to a Riemannian manifold and obtain necessary conditions for the fibers of and the base manifold to be Ricci soliton, almost Ricci soliton and Einstein. Moreover, we find necessary conditions for a vector field and its horizontal lift to be conformal on and respectively. Also, we calculate the scalar curvature of Ricci soliton . Finally, we obtain a necessary and sufficient condition for to be harmonic.

Ricci solitonsAlmost & η-Ricci solitonsScalar curvature

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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 2021, 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 Clairaut Riemannian maps (Kiran Meena and Akhilesh Yadav, 2021)2Meena 2021 On conformal collineation and almost Ricci solitons (Adara M. Blaga and Bang-Yen Chen, 2022)5Blaga 2022 Kähler-Ricci flow and conformal submersion (Hoan Nguyen, 2023)6Nguyen 2023 Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds (Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya, 2023)4Ghosh 2023 Clairaut conformal submersions from Ricci solitons (Murat Polat, 2023)1Polat 2023 On some 3-dimensional almost η-Ricci solitons with diagonal metrics (Adara M. Blaga, 2024)7Blaga 2024 Almost Gradient Ricci Solitons on Static Spacetime (Akhilesh Yadav and Tarun Saxena, 2025)3Yadav 2025 K-contact manifolds admitting some geometric solitons with Semi-Symmetric Non-Metric Connection (Bidhan Mondal, Nirabhra Basu and Arindam Bhattacharyya, 2026)8Mondal 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Clairaut conformal submersions from Ricci solitons

    Murat Polat PDF

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “conformal submersions”, “total manifolds”, “necessary conditions”

  2. Clairaut Riemannian maps

    Kiran Meena and Akhilesh Yadav PDF

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Same authors Kiran Meena, Akhilesh Yadav · Shared terms “necessary conditions”, “find necessary”, “sufficient conditions”

  3. Almost Gradient Ricci Solitons on Static Spacetime

    Akhilesh Yadav and Tarun Saxena PDF

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Akhilesh Yadav · Shared terms “base manifold”, “sufficient conditions”, “necessary”

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

    Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya PDF

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

  5. On conformal collineation and almost Ricci solitons

    Adara M. Blaga and Bang-Yen Chen PDF

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

  6. Kähler-Ricci flow and conformal submersion

    Hoan Nguyen PDF

    Shared terms “conformal submersions”, “sufficient conditions”, “necessary”

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

    Adara M. Blaga PDF

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

  8. K-contact manifolds admitting some geometric solitons with Semi-Symmetric Non-Metric Connection

    Bidhan Mondal, Nirabhra Basu and Arindam Bhattacharyya PDF

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “sufficient conditions”, “necessary”, “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)

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.