Paper · Jan 9, 2023

Generalized Ricci solitons and Einstein metrics on weak -contact manifolds

Vladimir Rovenski

math.DGarXiv:2301.03500v3
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Abstract

We study metric structures on a smooth manifold (introduced in our recent works and called a weak contact metric structure and a weak K-structure) which generalize the metric contact and K-contact structures, and allow a new look at the classical theory. First, we characterize weak K-contact manifolds among all weak contact metric manifolds by the property well known for K-contact manifolds, and find when a Riemannian manifold endowed with a unit Killing vector field forms a weak K-contact structure. Second, we find sufficient conditions for a weak K-contact manifold with parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.

Comments: 10 pages

Generalized & G₂ flowsRicci solitonsEinstein metricsSasakian & contact geometry

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    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Same author Vladimir Rovenski · Shared terms “contact structure”, “find sufficient”, “characterize weak”

  2. --Ricci solitons and Einstein metrics on a weak -Kenmotsu manifold

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    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Same author Vladimir Rovenski · Shared terms “classical theory”, “Ricci tensor”

  3. -Ricci solitons and -Einstein metrics on weak -Kenmotsu -manifolds

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    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Same author Vladimir Rovenski · Shared terms “metric structure”, “classical theory”, “soliton structure”

  4. Characterization of Sasakian Manifolds by Curvature Conditions

    Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra PDF

    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “weak contact”, “metric structure”, “characterize”

  5. --Ricci solitons on weak Kenmotsu -manifolds

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    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Same author Vladimir Rovenski · Shared terms “classical theory”, “Ricci tensor”

  6. Characterizations of weak almost -manifolds with curvature properties

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    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “metric structure”

  7. Sasaki with torsion manifolds and string backgrounds

    Beatrice Brienza, Anna Fino, Udhav Fowdar et al. PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Shared terms “metric structure”, “characterize”

  8. Geometric Flows of -Structures on 3-Sasakian 7-Manifolds

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    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Shared terms “parallel”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) 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

  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.