Paper · Dec 18, 2024

$η$ -Ricci solitons and $η$ -Einstein metrics on weak $β$ -Kenmotsu $f$ -manifolds

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Abstract

Recent interest among geometers in $f$ -structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by V. Rovenski and R. Wolak as a generalization of Hermitian and Kähler structures, as well as $f$ -structures, allow a fresh look at the classical theory. In this paper, we study a new $f$ -structure of this kind, called the weak $β$ -Kenmotsu $f$ -structure, as a generalization of K. Kenmotsu's concept. We prove that a weak $β$ -Kenmotsu $f$ -manifold is locally a twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $β = co n s t$ and equipped with an $η$ -Ricci soliton structure whose potential vector field satisfies certain conditions are $η$ -Einstein manifolds of constant scalar curvature.

Comments: 14 pages

Ricci solitons Einstein metrics Almost & η-Ricci solitons Kähler geometry Scalar curvature Sasakian & contact geometry

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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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

In the guidebook

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.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

$η$ -Ricci solitons and $η$ -Einstein metrics on weak $β$ -Kenmotsu $f$ -manifolds

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η-Ricci solitons and η-Einstein metrics on weak β-Kenmotsu f-manifolds

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$η$ -Ricci solitons and $η$ -Einstein metrics on weak $β$ -Kenmotsu $f$ -manifolds