Paper · Sep 14, 2017

Ricci Almost Solitons on semi-Riemannian Warped Products

Keti Tenenblat, Valter Borges

math.DGarXiv:1709.04604v2
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Abstract

We characterize Ricci almost solitons on semi-Riemannian warped products, considering the potential function to depend on the fiber or not. We show that the fiber is necessarily an Einstein manifold. As a consequence of our characterization we prove that when the potential function depends on the fiber, if the gradient of the warping function does not act by translations then the base and the warped product are also Einstein manifolds. Moreover, we show the existence of conformal vector fields on the base, the fiber and on the warped product. Assuming completeness of the warped product we provide a classification of such manifolds. When the potential function depends on the fiber and the gradient of the warping functions is an improper vector field, we show that the base is a Brinkmann space and the fiber is Ricci flat. We use the characterization also to prove that the potential function of a complete Ricci soliton depends only on the base.

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Submitted
Sep 14, 2017
Last revised
May 21, 2019 (v2)
Subjects
math.DG
DOI
10.1002/mana.201900242

Comments: 21 pages

Ricci solitonsEinstein metricsRicci-flat & Calabi–YauAlmost & η-Ricci solitonsRotational symmetry & warped products

Its neighbourhood

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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.

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 Destabilising compact warped product Einstein manifolds (Wafaa Batat, Stuart James Hall and Thomas Murphy, 2016)7Batat 2016 Classification of base of warped product almost Ricci solitons (José N. V. Gomes and Manoel V. M. Neto, 2017)3Gomes 2017 On warped product gradient η-Ricci solitons (Adara M. Blaga, 2017)5Blaga 2017 Gradient Estimates on Warped Product Gradient Almost Ricci Solitons (Willian Isao Tokura, Levi Adriano, Romildo Pina et al., 2019)8Tokura 2019 Characteristics of conformal Ricci soliton on warped product spaces (Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya, 2021)2Ganguly 2021 On noncompact warped product Ricci solitons (Valter Borges, 2022)1Borges 2022 A Characterization of Quasi-Einstein Metrics (Antonio Airton Freitas Filho, 2025)6Filho 2025 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)4Borges 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On noncompact warped product Ricci solitons

    Valter Borges

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons · Same author Valter Borges · Shared terms “warping function”

  2. Characteristics of conformal Ricci soliton on warped product spaces

    Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya

    Topics Almost & η-Ricci solitons, Ricci-flat & Calabi–Yau, Rotational symmetry & warped products · Shared terms “conformal vector”, “vector field”, “characterization”

  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, Einstein metrics · Shared terms “vector field”

  4. 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, Rotational symmetry & warped products, Ricci solitons · Same author Valter Borges · Shared terms “potential function”

  5. On warped product gradient -Ricci solitons

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “potential function”, “vector field”

  6. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “characterization”

  7. Destabilising compact warped product Einstein manifolds

    Wafaa Batat, Stuart James Hall and Thomas Murphy

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

  8. 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 “warping function”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… 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)

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.