Paper · May 11, 2023

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

Paritosh Ghosh, Hemangi Madhusudan Shah, Arindam Bhattacharyya

math.DGarXiv:2305.06597
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Abstract

In this research, we study the nature of -Einstein and gradient -Einstein soliton in the framework of almost coKähler manifolds and -almost coKähler manifolds. We find some expressions for scalar curvature of the almost coKähler manifold admitting -Einstein soliton in various cases. We also prove that if a -almost coKähler manifold admits a gradient -Einstein soliton, then either the manifold is coKähler, or -almost coKahler, or the soliton is trivial. We present an example which validates our results. Finally, we investigate the asymptotically harmonic manifolds admitting non-trivial Ricci solitons and show that they exhibit rigid behavior if for example, scalar curvature attains maximum.

Ricci solitonsAlmost & η-Ricci solitonsKähler geometryScalar curvature

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    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Same author Arindam Bhattacharyya · Shared terms “manifold admitting”

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    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Same author Arindam Bhattacharyya · Shared terms “manifold admitting”, “Einstein”

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    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Hemangi Madhusudan Shah · Shared terms “Einstein”

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

  5. Clairaut Riemannian maps

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

  6. Clairaut conformal submersions from Ricci solitons

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

  7. Investigations on a Riemannian manifold with a semi-symmetric non-metric connection and gradient solitons

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

  8. Characterizations of generalized Robertson-Walker spacetimes concerning gradient solitons

    Krishnendu De, Mohammad Nazrul Islam Khan and Uday Chand De PDF

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “Einstein 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 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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.