Kenmotsu metric as conformal η-Ricci soliton

Dipen Ganguly

Paper · Aug 26, 2021

Kenmotsu metric as conformal $η$ -Ricci soliton

Dipen Ganguly

math.DGarXiv:2108.11639

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Abstract

The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $η$ -Ricci soliton. Here, we have investigated the nature of the conformal $η$ -Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $η$ -Einstein Kenmotsu manifold admitting conformal $η$ -Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $η$ -Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $η$ -Ricci soliton on Kenmotu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal $η$ -Ricci soliton on Kenmotsu manifold.

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Submitted: Aug 26, 2021
Subjects: math.DG, math-ph

Comments: 16 pages

Ricci solitons Almost & η-Ricci solitons Sasakian & contact geometry

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Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “Kenmotsu”, “Einstein”
$η$ -Ricci solitons and $η$ -Einstein metrics on weak $β$ -Kenmotsu $f$ -manifolds
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Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “Kenmotsu”, “Reeb vector”, “vector field”
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Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “admits conformal”, “admitting conformal”, “vector field”
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Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “manifold admitting”, “vector field”, “relation”
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Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “Kenmotsu”, “vector field”, “Reeb”
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Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “vector field”, “potential vector”

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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) Sasakian & contact geometry Sasakian manifolds are odd-dimensional manifolds whose metric cone is Kähler. They carry a transverse Kähler structure, so Kähler–Ricci flow has an… Start with: THE SASAKI–RICCI FLOW (SMOCZYK et al., 2010)

In the guidebook

11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[1] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

Kenmotsu metric as conformal $η$ -Ricci soliton

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