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Journal ArticleDOI

Conformally flat contact metric manifolds

01 Apr 2001-Journal of Geometry (Springer Science and Business Media LLC)-Vol. 70, Iss: 1, pp 66-76
TL;DR: In this article, a couple of classes of conformally flat contact metric manifolds have been classified and characterized as hypersurfaces of 4-dimensional Kaehler Einstein (in particular, Calabi-Yau) manifolds.
Abstract: A couple of classes of conformally flat contact metric manifolds have been classified. Conformally flat contact manifolds have been characterized as hypersurfaces of 4-dimensional Kaehler Einstein (in particular, Calabi-Yau) manifolds.
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Journal ArticleDOI
TL;DR: In this article, the authors considered φ conformally at, conharmonically at, projectively at and concircularly at Lorentzian α Sasakian manifolds.
Abstract: In this study we consider φ conformally at, φ conharmonically at, φ projectively at and φ concircularly at Lorentzian α Sasakian manifolds. In all cases, we get the manifold will be an η Einstein manifold.

18 citations

01 Jan 2013
TL;DR: In this paper, the object of the present paper is to study x -projectively flat and f -projective flat 3-dimensional normal almost contact metric manifolds, and an illustrative example is given.
Abstract: The object of the present paper is to study x -projectively flat and f -projectively flat 3-dimensional normal almost contact metric manifolds. An illustrative example is given.

17 citations

Journal ArticleDOI
TL;DR: In this article, a Ricci soliton with potential vector field V collinear with ξ at each point under different curvature conditions was studied on a contact metric manifold M2n+1(ϕ, ξ, η, g).
Abstract: We study on a contact metric manifold M2n+1(ϕ, ξ, η, g) such that g is a Ricci soliton with potential vector field V collinear with ξ at each point under different curvature conditions: (i) M is of pointwise constant ξ-sectional curvature, (ii) M is conformally flat.

16 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian.
Abstract: In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying \({Q\varphi = \varphi Q}\) is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, fη) are presented.

14 citations


Cites result from "Conformally flat contact metric man..."

  • ...Generalizing a result of Blair and Koufogiorgos [2], the authors, Koufogiorgos and Sharma [ 8 ] proved that a conformally flat contact metric manifold satisfying Qξ = ( Trl )ξ and K (ξ, X )+K (ξ, ϕ X )independentof X ⊥ξ isofconstantcurvature.Recentlyin[10],Gouli-Andreou andTsolakidouprovedthatthehypothesisonthesectionalcurvatureis redundant and prove that a conformally flat contact metric manifold with Qξ = ( Trl )ξ is of constant ......

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Book ChapterDOI
01 Jan 2005
TL;DR: The authors surveys a number of results and open questions concerning the curvature of Riemannian metrics associated to a contact form, including the curvatures of contact forms and contact boxes.
Abstract: This essay surveys a number of results and open questions concerning the curvature of Riemannian metrics associated to a contact form.

11 citations


Cites background from "Conformally flat contact metric man..."

  • ...Then in [21] G. Calvaruso, D. Perrone and L. Vanhecke showed that i nd imension 3 the only conformally flat contact metric structures, for which ξ i sa n e igenvector of the Ricci operator, are those of constant curvature 0 or 1. An attempt was made in [ 24 ] to generaliz et hi st o higher dimensions by assuming another condition in addition to ξ being an eigenvector of the Ricci operator....

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  • ...Recently Ghosh, Koufogiorgos and Sharma [ 24 ] have shown that a conformally flat contact metric manifold of dimension ≥ 5w ith a strongly pseudo-convex integrable CR-structure is of constant curvature +1. As we have seen, i nd imension ≥ 5, a contact metric structure of constant curvature must be of constant curvature +1 and is Sasakian; and i nd imension 3, a contact metric structure of constant curvature must be of constant curvature 0 or ......

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