On non-gradient (m,ρ)-quasi-Einstein contact metric manifolds

Dhriti Sundar Patra; Vladimir Rovenski

Paper · Oct 28, 2020

On non-gradient $(m, ρ)$ -quasi-Einstein contact metric manifolds

Dhriti Sundar Patra, Vladimir Rovenski

math.DGarXiv:2010.15150

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Abstract

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m, ρ)$ -quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$ -contact or Sasakian manifold $M^{2 n + 1}$ admits a closed $(m, ρ)$ -quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2 n (2 n + 1)$ , and for the particular case – a non-Sasakian $(k, μ)$ -contact structure – it is locally isometric to the product of a Euclidean space $\RR^{n + 1}$ and a sphere $S^{n}$ of constant curvature $4$ . Next, we prove that if a compact contact or $H$ -contact metric manifold admits an $(m, ρ)$ -quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$ -contact $η$ -Einstein manifold.

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Submitted: Oct 28, 2020
Subjects: math.DG

Comments: 12 pages

Ricci solitons Einstein metrics Almost & η-Ricci solitons 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) 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

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.

[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.

On non-gradient $(m, ρ)$ -quasi-Einstein contact metric manifolds

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