Paper · Aug 24, 2021

-Ricci-Yamabe Soliton and Contact Geometry

Dibakar Dey

As on arXiv

Abstract

It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the -Ricci soliton, the notion of -Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A -dimensional non-Sasakian -contact metric manifold admitting -Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold admits -Ricci-Yamabe soliton under certain conditions on the soliton vector field , then is -Ricci flat, positive Sasakian and the transverse geometry of is Fano. In addition, the Sasakian 3-metric is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field is an infinitesimal automorphism of the contact metric structure.

Submitted
Aug 24, 2021
Subjects
math.DG

Comments: 16 pages

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  1. Ricci-Yamabe solitons on the Lie group

    Abdou Bousso and Ameth Ndiaye

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “Ricci Yamabe”, “Yamabe soliton”, “vector field”

  2. Complete gradient Einstein-type Sasakian manifolds with

    Shun Maeta

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “unit sphere”, “Yamabe soliton”

  3. Characterization of Sasakian Manifolds by Curvature Conditions

    Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra

    Topics Sasakian & contact geometry · Shared terms “metric structure”

  4. Sasakian lift of Kaehler manifold and -Sasakian Ricci solitons

    Piotr Dacko

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Kähler geometry · Shared terms “metric structure”, “vector field”, “automorphisms”

  5. Geometry of para-Sasakian metric as an almost conformal -Ricci soliton

    Sumanjit Sarkar, Santu Dey and Arindam Bhattacharyya

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, 3-manifolds · Shared terms “soliton vector”, “vector field”, “manifold admitting”

  6. K-contact manifolds admitting some geometric solitons with Semi-Symmetric Non-Metric Connection

    Bidhan Mondal, Nirabhra Basu and Arindam Bhattacharyya

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “Ricci Yamabe”, “Yamabe soliton”, “manifold admitting”

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

    Vladimir Rovenski

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci-flat & Calabi–Yau · Shared terms “vector field”

  8. Kenmotsu metric as conformal -Ricci soliton

    Dipen Ganguly

    Topics Sasakian & contact geometry, Almost & η-Ricci solitons, Ricci solitons · Shared terms “vector field”, “potential vector”, “manifold admitting”

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.

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.