Paper · Jan 3, 2020

On a Class of Gradient Almost Ricci Solitons

Sinem Güler

math.DGarXiv:2001.00749
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Abstract

In this study, we provide some classifications for half-conformally flat gradient -almost Ricci solitons, denoted by , in both Lorentzian and neutral signature. First, we prove that if is a non-zero constant, then is locally isometric to a warped product of the form , where and is of constant sectional curvature. On the other hand, if , then it is locally a Walker manifold. Then, we construct an example of 4-dimensional steady gradient -almost Ricci solitons in neutral signature. At the end, we give more physical applications of gradient Ricci solitons endowed with the standard static spacetime metric.

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Submitted
Jan 3, 2020
Subjects
math.DG
Ricci solitonsAlmost & η-Ricci solitons4-manifoldsRotational symmetry & warped productsPhysics

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Classification of base of warped product almost Ricci solitons (José N. V. Gomes and Manoel V. M. Neto, 2017)3Gomes 2017 Basic structural equations for almost Ricci-harmonic solitons and applications (Abimbola Abolarinwa, 2018)4Abolarinwa 2018 Gradient Estimates on Warped Product Gradient Almost Ricci Solitons (Willian Isao Tokura, Levi Adriano, Romildo Pina et al., 2019)1Tokura 2019 Perfect fluid spacetimes and gradient solitons (Krishnendu De Uday Chand De, Abdallah Abdelhameed Syied, Nasser Bin Turki et al., 2021)7De 2021 Back to almost Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2023)8Rovenski 2023 Almost Ricci solitons on Finsler spaces (Qiaoling Xia, 2024)5Xia 2024 Local structure of gradient almost Ricci solitons with harmonic Weyl tensor (Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos, 2025)2Borges 2025 Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity (Mohammadjavad Habibivostakolaei, Abbas M. Sherif and Yen-Kheng Lim, 2026)6Habibivostak 2026
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  1. Gradient Estimates on Warped Product Gradient Almost Ricci Solitons

    Willian Isao Tokura, Levi Adriano, Romildo Pina et al.

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

  2. Local structure of gradient almost Ricci solitons with harmonic Weyl tensor

    Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos

    Topics Almost & η-Ricci solitons, 4-manifolds, Rotational symmetry & warped products · Shared terms “gradient almost”

  3. Classification of base of warped product almost Ricci solitons

    José N. V. Gomes and Manoel V. M. Neto

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

  4. Basic structural equations for almost Ricci-harmonic solitons and applications

    Abimbola Abolarinwa

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “gradient almost”

  5. Almost Ricci solitons on Finsler spaces

    Qiaoling Xia

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “gradient almost”, “gradient Ricci”

  6. Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity

    Mohammadjavad Habibivostakolaei, Abbas M. Sherif and Yen-Kheng Lim

    Topics Almost & η-Ricci solitons, 4-manifolds, Rotational symmetry & warped products · Shared terms “signature”, “conformally flat”, “Lorentzian”

  7. Perfect fluid spacetimes and gradient solitons

    Krishnendu De Uday Chand De, Abdallah Abdelhameed Syied, Nasser Bin Turki et al.

    Topics Almost & η-Ricci solitons, 4-manifolds, Physics · Shared terms “Walker”, “Lorentzian”, “gradient Ricci”

  8. Back to almost Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Almost & η-Ricci solitons, Ricci solitons

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) 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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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

  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.