Paper · Feb 23, 2017

Realizations of some contact metric manifolds as Ricci soliton real hypersurfaces

Jong Taek Cho, Takahiro Hashinaga, Akira Kubo, Yuichiro Taketomi, Hiroshi Tamaru

math.DGarXiv:1702.07256
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Abstract

Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case.These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces in noncompact real two-plane Grassmannians. These realizations enable us to prove, in a Lie-theoretic way, that all of them are actually Ricci soliton.

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Submitted
Feb 23, 2017
Subjects
math.DG
DOI
10.1016/j.geomphys.2017.08.013

Comments: 21 pages

Ricci solitons3-manifoldsHomogeneous spacesBoundaries & non-compactSasakian & contact geometry

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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 Certain results on Kenmotsu pseudo-metric manifolds (Devaraja Mallesha Naik, Venkatesha and D. G. Prakasha, 2018)5Naik 2018 N(k)-contact metric as ∗-conformal Ricci soliton (Dibakar Dey and Pradip Majhi, 2020)3Dey 2020 Remarks on Riemann and Ricci solitons in (α,β)-contact metric manifolds (Adara M. Blaga and Dan Radu Latcu, 2020)1Blaga 2020 On non-gradient (m,ρ)-quasi-Einstein contact metric manifolds (Dhriti Sundar Patra and Vladimir Rovenski, 2020)8Patra 2020 A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups (Yuji Kondo and Hiroshi Tamaru, 2020)2Kondo 2020 Codimension one Ricci soliton subgroups of solvable Iwasawa groups (Miguel Dominguez-Vazquez, Victor Sanmartin-Lopez and Hiroshi Tamaru, 2021)4Dominguez-Va 2021 Nilpotent Lie algebras obtained by quivers and Ricci solitons (Fumika Mizoguchi and Hiroshi Tamaru, 2024)6Mizoguchi 2024 Characterization of Sasakian Manifolds by Curvature Conditions (Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra, 2026)7Nayak 2026
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  1. Remarks on Riemann and Ricci solitons in -contact metric manifolds

    Adara M. Blaga and Dan Radu Latcu

    Topics Sasakian & contact geometry, Ricci solitons

  2. A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups

    Yuji Kondo and Hiroshi Tamaru

    Topics 3-manifolds, Homogeneous spaces, Ricci solitons · Same author Hiroshi Tamaru · Shared terms “Lie groups”

  3. -contact metric as -conformal Ricci soliton

    Dibakar Dey and Pradip Majhi

    Topics Sasakian & contact geometry, Ricci solitons

  4. Codimension one Ricci soliton subgroups of solvable Iwasawa groups

    Miguel Dominguez-Vazquez, Victor Sanmartin-Lopez and Hiroshi Tamaru

    Topics Homogeneous spaces, Boundaries & non-compact, Ricci solitons · Same author Hiroshi Tamaru · Shared terms “Lie groups”, “simply connected”

  5. Certain results on Kenmotsu pseudo-metric manifolds

    Devaraja Mallesha Naik, Venkatesha and D. G. Prakasha

    Topics Sasakian & contact geometry, 3-manifolds, Ricci solitons

  6. Nilpotent Lie algebras obtained by quivers and Ricci solitons

    Fumika Mizoguchi and Hiroshi Tamaru

    Topics Homogeneous spaces, Ricci solitons · Same author Hiroshi Tamaru · Shared terms “Lie algebras”, “Lie groups”, “simply connected”

  7. Characterization of Sasakian Manifolds by Curvature Conditions

    Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra

    Topics Sasakian & contact geometry · Shared terms “nullity condition”, “underlying”

  8. On non-gradient -quasi-Einstein contact metric manifolds

    Dhriti Sundar Patra and Vladimir Rovenski

    Topics Sasakian & contact geometry, Ricci solitons

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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