Ricci ρ-solitons on 3-dimensional η-Einstein almost Kenmotsu mandifolds

Sh. Azami; Gh. Fasihi-Ramandi

Paper · Feb 20, 2019

Ricci $ρ$ -solitons on 3-dimensional $η$ -Einstein almost Kenmotsu mandifolds

Sh. Azami, Gh. Fasihi-Ramandi

math.DGarXiv:1902.08176

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Abstract

In this paper the notion of Ricci $ρ$ -soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3- dimensional almost Kenmotsu Einstein manifold $M$ be a $ρ$ -soliton, then $M$ is a Kenmotsu manifold of constant sectional curvature $- 1$ and the $ρ$ -soliton is expanding, with $λ = 2$ .

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Submitted: Feb 20, 2019
Subjects: math.DG

Ricci flow Ricci solitons Einstein metrics 3-manifolds 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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
10A.2
Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.
10A.4
Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization
Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.

[1] 10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

[2] 10A.2
Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

[3] 10A.4
Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization
Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.

Ricci $ρ$ -solitons on 3-dimensional $η$ -Einstein almost Kenmotsu mandifolds

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