Almost Kenmotsu 3-manifolds satisfying some generalized nullity conditions

In this paper, a three-dimensional almost Kenmotsu manifold M3 satisfying the generalized (k,?)'-nullity condition is investigated. We mainly prove that on M3 the following statements are equivalent: (1) M3 is ?-symmetric; (2) the Ricci tensor of M3 is cyclic-parallel; (3) the Ricci tensor of M3 is of Codazzi type; (4) M3 is conformally flat with scalar curvature invariant along the Reeb vector field; (5) M3 is locally isometric to either the hyperbolic space H3(-1) or the Riemannian product H2(-4) x R.

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Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold

Carpathian Mathematical Publications ◽

10.15330/cmp.11.1.59-69 ◽

2019 ◽

Vol 11 (1) ◽

pp. 59-69 ◽

Cited By ~ 2

Author(s):

A. Ghosh

Keyword(s):

Vector Field ◽

Scalar Curvature ◽

Ricci Soliton ◽

Kenmotsu Manifold ◽

Curvature Invariant ◽

Ricci Operator ◽

Reeb Vector ◽

Reeb Vector Field

First, we prove that if the Reeb vector field $\xi$ of a Kenmotsu manifold $M$ leaves the Ricci operator $Q$ invariant, then $M$ is Einstein. Next, we study Kenmotsu manifold whose metric represents a Ricci soliton and prove that it is expanding. Moreover, the soliton is trivial (Einstein) if either (i) $V$ is a contact vector field, or (ii) the Reeb vector field $\xi$ leaves the scalar curvature invariant. Finally, it is shown that if the metric of a Kenmotsu manifold represents a gradient Ricci almost soliton, then it is $\eta$-Einstein and the soliton is expanding. We also exhibited some examples of Kenmotsu manifold that admit Ricci almost solitons.

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Static Perfect Fluid Space-Time on Almost Kenmotsu Manifolds

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-61-2021-41-51 ◽

2021 ◽

Vol 61 ◽

pp. 41-51

Author(s):

Huchchappa A. Kumara ◽

◽

Venkatesha Venkatesha ◽

Devaraja M. Naik

Keyword(s):

Vector Field ◽

Perfect Fluid ◽

Space Time ◽

Kenmotsu Manifold ◽

Curvature Invariant ◽

Reeb Vector ◽

Reeb Vector Field ◽

Kenmotsu Manifolds ◽

Almost Kenmotsu Manifolds ◽

Spatial Factor

In this work, we intend to investigate the characteristics of static perfect fluid space-time metrics on almost Kenmotsu manifolds. At first we prove that if a Kenmotsu manifold $M$ is the spatial factor of static perfect fluid space-time then it is $\eta$-Einstein. Moreover, if the Reeb vector field $\xi$ leaves the scalar curvature invariant, then $M$ is Einstein. Next we consider static perfect fluid space-time on almost Kenmotsu $(\kappa,\mu)'$-manifolds and give some characteristics under certain conditions.

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Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

Canadian Mathematical Bulletin ◽

10.4153/cmb-2017-049-5 ◽

2018 ◽

Vol 61 (3) ◽

pp. 543-552

Author(s):

Imsoon Jeong ◽

Juan de Dios Pérez ◽

Young Jin Suh ◽

Changhwa Woo

Keyword(s):

Vector Field ◽

Differential Operator ◽

Ricci Tensor ◽

Real Hypersurface ◽

Real Hypersurfaces ◽

Reeb Vector ◽

Reeb Vector Field ◽

Lie Derivatives ◽

Lie Derivation

AbstractOn a real hypersurface M in a complex two-plane Grassmannian G2() we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka–Webster connection . We give a classification of real hypersurfaces M on G2() satisfying , where ξ is the Reeb vector field on M and S the Ricci tensor of M.

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The k-almost Yamabe solitons and almost coKähler manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821501796 ◽

2021 ◽

pp. 2150179

Author(s):

Xiaomin Chen ◽

Uday Chand De

Keyword(s):

Vector Field ◽

Riemannian Product ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Almost Kähler Manifold ◽

Almost Kähler ◽

Yamabe Solitons ◽

Potential Vector Field

In this paper, we study almost coKähler manifolds admitting [Formula: see text]-almost Yamabe solitons [Formula: see text]. First, we obtain a classification of almost coKähler [Formula: see text]-manifolds admitting nontrivial closed [Formula: see text]-almost Yamabe solitons. Next, we consider an almost [Formula: see text]-coKähler manifold admitting a nontrivial [Formula: see text]-almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field [Formula: see text] is collinear with the Reeb vector field. For the potential vector field [Formula: see text] being orthogonal to the Reeb vector field, we also obtain two results.

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Curvature of K-contact Semi-Riemannian Manifolds

Canadian Mathematical Bulletin ◽

10.4153/cmb-2013-016-7 ◽

2014 ◽

Vol 57 (2) ◽

pp. 401-412 ◽

Cited By ~ 7

Author(s):

Domenico Perrone

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Sectional Curvature ◽

Riemannian Manifolds ◽

Lorentzian Manifold ◽

Constant Sectional Curvature ◽

Conformally Flat ◽

Reeb Vector ◽

Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

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A study of three-dimensional paracontact (κ̃,μ̃,ν̃)-spaces

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817501067 ◽

2017 ◽

Vol 14 (07) ◽

pp. 1750106 ◽

Cited By ~ 5

Author(s):

İrem Küpeli Erken ◽

Cengizhan Murathan

Keyword(s):

Vector Field ◽

Three Dimensional ◽

Reeb Vector ◽

Reeb Vector Field

This paper is a study of three-dimensional paracontact metric [Formula: see text]-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field [Formula: see text] is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric [Formula: see text]-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases [Formula: see text] [Formula: see text] and construct new examples of such manifolds for each case. We also show the existence of paracontact metric [Formula: see text] spaces with dimension greater than 3, such that [Formula: see text] but [Formula: see text]

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Certain homogeneous paracontact three-dimensional Lorentzian metrics

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500067 ◽

2017 ◽

Vol 11 (01) ◽

pp. 1850006

Author(s):

Ali Haji-Badali ◽

Elham Sourchi

Keyword(s):

Vector Field ◽

Three Dimensional ◽

Einstein Manifolds ◽

Reeb Vector ◽

Reeb Vector Field ◽

Lorentzian Metrics

In this paper, we study three-dimensional homogeneous paracontact metric manifolds for which the Reeb vector field of the underlying paracontact structure satisfies a nullity condition. We give example of paraSasakian and non-paraSasakian [Formula: see text]-manifolds. Finally, we exhibit explicit example of [Formula: see text]-Einstein manifolds.

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Almost $$*$$-Ricci soliton on paraKenmotsu manifolds

Arabian Journal of Mathematics ◽

10.1007/s40065-019-00269-7 ◽

2019 ◽

Vol 9 (3) ◽

pp. 715-726 ◽

Cited By ~ 1

Author(s):

V. Venkatesha ◽

H. Aruna Kumara ◽

Devaraja Mallesha Naik

Keyword(s):

Vector Field ◽

Constant Curvature ◽

Three Dimensional ◽

Ricci Soliton ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Almost Ricci Soliton ◽

Potential Vector Field

Abstract We consider almost $$*$$ ∗ -Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of $$\eta $$ η -Einstein paraKenmotsu manifold is $$*$$ ∗ Ricci soliton, then M is Einstein. Next, we show that if $$\eta $$ η -Einstein paraKenmotsu manifold admits a gradient almost $$*$$ ∗ -Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field $$\xi $$ ξ . Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature $$-1$$ - 1 . An illustrative example is given to support the obtained results.

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N(k)-paracontact three metric as a Eta-Ricci soliton

SERIES III - MATEMATICS, INFORMATICS, PHYSICS ◽

10.31926/but.mif.2020.13.62.2.16 ◽

2021 ◽

Vol 13(62) (2) ◽

pp. 581-594

Author(s):

Debabrata Kar ◽

Pradip Majhi

Keyword(s):

Vector Field ◽

Killing Vector ◽

Three Dimensional ◽

Ricci Soliton ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Infinitesimal Automorphism ◽

Potential Vector Field

In this paper, we study Eta-Ricci soliton (η-Ricci soliton) on three dimensional N(k)-paracontact metric manifolds. We prove that the scalar curvature of an N(k)-paracontact metric manifold admitting η-Ricci solitons is constant and the manifold is of constant curvature k. Also, we prove that such manifolds are Einstein. Moreover, we show the condition of that the η-Ricci soliton to be expanding, steady or shrinking. In such a case we prove that the potential vector field is Killing vector field. Also, we show that the potential vector field is an infinitesimal automorphism or it leaves the structure tensor in the direction perpendicular to the Reeb vector field ξ. Finally, we illustrate an example of a three dimensional N(k)-paracontact metric manifold admitting an η-Ricci soliton

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Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

Complex Manifolds ◽

10.1515/coma-2017-0005 ◽

2017 ◽

Vol 4 (1) ◽

pp. 43-72 ◽

Cited By ~ 1

Author(s):

Martin de Borbon

Keyword(s):

Vector Field ◽

Einstein Metrics ◽

Projective Line ◽

Tangent Cones ◽

Reeb Vector ◽

Reeb Vector Field ◽

Complex Curves ◽

Complex Dimensions ◽

Irregular Case ◽

Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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