Paper · Jun 30, 2023

Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

Roberto Mossa, Giovanni Placini

math.DGarXiv:2306.17783Ann. Global Anal. Geom. (2024) 65:8
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Abstract

We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, -Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler-Ricci soliton on which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds.

Ricci solitonsKähler–Ricci solitonsSphere theoremsKähler geometryHomogeneous spacesSasakian & contact geometry

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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 2021, 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 Holomorphic isometries into homogeneous bounded domains (Andrea Loi and Roberto Mossa, 2022)3Loi 2022 Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven (Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al., 2022)4Chang 2022 Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds (A. Loi and R. Mossa, 2023)1Loi 2023 Ricci iterations of well-behaved Kähler metrics (Andrea Loi and Giovanni Placini, 2023)7Loi 2023 Transverse Rigidity of Shrinking Sasaki-Ricci Solitons (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2025)8Chang 2025 Generalized η-Ricci solitons on LP-Sasakian manifolds admitting the general connection (Murat Altunbaş and Ayşe Karanlık Akpınar, 2025)6Altunbaş 2025 Universal embeddings of flag manifolds and rigidity phenomena (Andrea Loi, Roberto Mossa and Fabio Zuddas, 2025)2Loi 2025 Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons (Shu-Cheng Chang, Yingbo Han and Chin-Tung Wu, 2025)5Chang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa PDF

    Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same author Roberto Mossa · Shared terms “holomorphic isometries”, “homogeneous bounded”, “flag manifold”

  2. Universal embeddings of flag manifolds and rigidity phenomena

    Andrea Loi, Roberto Mossa and Fabio Zuddas PDF

    Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same author Roberto Mossa · Shared terms “flag manifold”, “homogeneous bounded”, “bounded domain”

  3. Holomorphic isometries into homogeneous bounded domains

    Andrea Loi and Roberto Mossa PDF

    Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same author Roberto Mossa · Shared terms “homogeneous bounded”, “holomorphic isometries”, “bounded domain”

  4. Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

    Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al. PDF

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler geometry · Shared terms “Sasaki Ricci”

  5. Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Yingbo Han and Chin-Tung Wu PDF

    Topics Sasakian & contact geometry, Sphere theorems, Kähler–Ricci solitons · Shared terms “Sasaki Ricci”, “holomorphic”

  6. Generalized Ricci solitons on LP-Sasakian manifolds admitting the general connection

    Murat Altunbaş and Ayşe Karanlık Akpınar PDF

    Topics Sasakian & contact geometry, Ricci solitons

  7. Ricci iterations of well-behaved Kähler metrics

    Andrea Loi and Giovanni Placini PDF

    Topics Sphere theorems, Kähler–Ricci solitons, Kähler geometry · Same author Giovanni Placini · Shared terms “space forms”, “Einstein”

  8. Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “Sasaki Ricci”, “Einstein”

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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.