Paper · Mar 23, 2025

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

Vladimir Rovenski

math.DGarXiv:2503.18031v5
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Abstract

Weak almost contact metric manifolds (i.e., the complex structure is replaced by a nonsingular skew-symmetric tensor), defined by the author and R. Wolak, allow a new look at the classical theory and find novel applications. An important case of these manifolds, which is locally a twisted product, is a weak -Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the -Ricci tensor is adapted to weak almost contact manifolds, the interaction of the --Ricci soliton with the weak -Kenmotsu structure (with ) is studied and new characteristics of Einstein metrics are obtained.

Comments: 14 pages

Ricci solitonsEinstein metricsAlmost & η-Ricci solitonsSasakian & contact geometry

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  1. --Ricci solitons on weak Kenmotsu -manifolds

    Vladimir Rovenski PDF

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Einstein metrics · Same author Vladimir Rovenski · Shared terms “Kenmotsu”, “manifold defined”, “twisted product”

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

    Vladimir Rovenski PDF

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Einstein metrics · Same author Vladimir Rovenski · Shared terms “Kenmotsu”, “twisted product”, “classical theory”

  3. Generalized Ricci solitons and Einstein metrics on weak -contact manifolds

    Vladimir Rovenski PDF

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

  4. Einstein-type metrics and generalized Ricci solitons on weak -K-contact manifolds

    Vladimir Rovenski PDF

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Einstein metrics · Same author Vladimir Rovenski · Shared terms “nonsingular skew”, “symmetric tensor”, “complex structure”

  5. Kenmotsu metric as conformal -Ricci soliton

    Dipen Ganguly PDF

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “Kenmotsu manifold”, “Einstein”

  6. Characterizations of weak almost -manifolds with curvature properties

    Sourav Nayak, Dhriti Sundar Patra and Vladimir Rovenski PDF

    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “weak almost”, “almost contact”, “Wolak”

  7. Back to almost Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Same author Vladimir Rovenski · Shared terms “Ricci tensor”

  8. -Ricci Solitons on Kenmotsu 3-Manifolds

    K. De and U. C. De PDF

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “Kenmotsu”, “Ricci tensor”

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) 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) 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.