Classification of base of warped product almost Ricci solitons

José N. V. Gomes; Manoel V. M. Neto

Paper · Apr 11, 2017

Classification of base of warped product almost Ricci solitons

José N. V. Gomes, Manoel V. M. Neto

math.DGarXiv:1704.03130v2

View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper we study a Ricci-Hessian type manifold $(M, g, φ, f, λ)$ which is closely related to the construction of almost Ricci soliton realized as a warped product. We classify certain classes of the Ricci-Hessian type manifolds and derive some implications for almost Ricci solitons and generalized $m$ –quasi-Einstein manifolds. We consider two complementary cases: $\nabla f$ and $\nabla φ$ are linearly independent in $C^{\infty} (M)$ –module $X (M)$ ; and $\nabla f = h \nabla φ$ for a smooth function $h$ on $M$ . In the first case we show that the vector field $\nabla λ$ belongs to the $C^{\infty} (M)$ –module generated by $\nabla f$ and $\nabla φ$ , while in the second case, under additional hypothesis, the manifold is, around any regular point of $f$ , locally isometric to a warped product.

View PDF View HTML arXiv page

Submitted: Apr 11, 2017
Last revised: Aug 11, 2017 (v2)
Subjects: math.DG

Comments: 18 pages

Ricci solitons Einstein metrics Almost & η-Ricci solitons Rotational symmetry & warped products

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

A note on gradient Ricci soliton warped metrics
José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro2021

Topics Rotational symmetry & warped products, Ricci solitons · Same author José N. V. Gomes · Shared terms “Hessian type”, “Ricci Hessian”, “type manifolds”
Destabilising compact warped product Einstein manifolds
Wafaa Batat, Stuart James Hall and Thomas Murphy2016

Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “quasi Einstein”
Ricci Almost Solitons on semi-Riemannian Warped Products
Keti Tenenblat and Valter Borges2017

Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “vector field”
Back to almost Ricci solitons
Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok2023

Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons
Gradient Estimates on Warped Product Gradient Almost Ricci Solitons
Willian Isao Tokura, Levi Adriano, Romildo Pina et al.2019

Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons
On warped product gradient $η$ -Ricci solitons
Adara M. Blaga2017

Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “vector field”
Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow
Atreyee Bhattacharya and Sayoojya Prakash2025

Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “quasi Einstein”
On a Class of Gradient Almost Ricci Solitons
Sinem Güler2020

Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “locally isometric”

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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.

Classification of base of warped product almost Ricci solitons

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters