Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

Giovanni Placini

Paper · Feb 4, 2021

Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

Giovanni Placini

math.DGarXiv:2102.02523

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Abstract

Let $(g, X)$ be a Sasaki-Ricci soliton on a Sasakian manifold $S$ . We prove that if $(S, g)$ admits a local Sasakian immersion in a Sasakian space form $S (N, c)$ of constant $φ$ -sectional curvature $c$ , then $S$ is $η$ -Einstein and its $η$ -Einstein constants are rational. Moreover, if $c \leq - 3$ , $S$ is locally equivalent to the Sasakian space form $S (n, c)$ and its $η$ -Einstein constants are determined by $c$ . Further results are obtained in the compact setting, i.e. when $c > - 3$ , under additional hypotheses.

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Submitted: Feb 4, 2021
Subjects: math.DG, math.MG
Journal: Journal of Geometry and Physics, 166, 104265, (2021)
DOI: 10.1016/j.geomphys.2021.104265

Comments: 9 pages, comments are welcome

Ricci solitons Sphere theorems 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 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) 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) Sasakian & contact geometry Sasakian manifolds are odd-dimensional manifolds whose metric cone is Kähler. They carry a transverse Kähler structure, so Kähler–Ricci flow has an… Start with: THE SASAKI–RICCI FLOW (SMOCZYK et al., 2010)

In the guidebook

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.
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] 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] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[3] 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.

Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

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