Certain homogeneous paracontact three-dimensional Lorentzian metrics

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.

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

2017 ◽  
Vol 14 (07) ◽  
pp. 1750106 ◽  
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]

scholarly journals Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces T1,1 and Yp,q

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 591
Author(s):  
Mihai Visinescu
Keyword(s):  
Vector Field ◽  
Hamiltonian Systems ◽  
Ricci Flow ◽  
Contact Structures ◽  
Contact Form ◽  
Einstein Manifolds ◽  
Integrable Hamiltonian Systems ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Completely Integrable Hamiltonian Systems

In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry. We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form. We examine the modifications of the action–angle coordinates by the Sasaki–Ricci flow. We then pass to the particular cases of the contact structures of the five-dimensional Sasaki–Einstein manifolds T1,1 and Yp,q.

scholarly journals Almost Kenmotsu 3-manifolds satisfying some generalized nullity conditions

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 197-206
Author(s):  
Wenjie Wang ◽  
Ximin Liu
Keyword(s):  
Vector Field ◽  
Ricci Tensor ◽  
Three Dimensional ◽  
Conformally Flat ◽  
Kenmotsu Manifold ◽  
Curvature Invariant ◽  
Riemannian Product ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Codazzi Type

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.

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

2019 ◽  
Vol 9 (3) ◽  
pp. 715-726 ◽  
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.

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

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

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

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
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.

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

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.

Ricci almost solitons and contact geometry

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Vector Field ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Jacobi Field ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Rigidity Results ◽  
Parallel Ricci Tensor ◽  
Potential Vector Field

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

scholarly journals B.-Y. Chen inequalities for semislant submanifolds in Sasakian space forms

2003 ◽  
Vol 2003 (27) ◽  
pp. 1731-1738 ◽  
Author(s):  
Dragoş Cioroboiu
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Sasakian Manifold ◽  
Sharp Inequality ◽  
Space Forms ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Sasakian Space Forms ◽  
Riemannian Space Forms

Chen (1993) established a sharp inequality for the sectional curvature of a submanifold in Riemannian space forms in terms of the scalar curvature and squared mean curvature. The notion of a semislant submanifold of a Sasakian manifold was introduced by J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, and M. Fernandez (1999). In the present paper, we establish Chen inequalities for semislant submanifolds in Sasakian space forms by using subspaces orthogonal to the Reeb vector fieldξ.

scholarly journals Reeb periodic orbits after a bypass attachment

2013 ◽  
Vol 35 (2) ◽  
pp. 615-672
Author(s):  
ANNE VAUGON
Keyword(s):  
Periodic Orbits ◽  
Contact Structure ◽  
Convex Surface ◽  
Three Dimensional ◽  
Contact Manifold ◽  
Contact Structures ◽  
Contact Homology ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Manifold With Boundary

AbstractOn a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

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