Abstract:The present paper is devoted to study the curvature and torsion of Frenet Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non-Frenet Legendre curves (with null tangents or null normals or null binormals) are obtained. Many examples of Legendre curves are constructed.Some of the present results are analogous to those obtained by the author in  for Frenet Legendre curves in 3-dimensional normal almost contact manifolds.
Mathemat… Show more
“…A semi-Riemannian metric g is said to be an associated metric if there exists a tensor ϕ of type (1,1), such that…”
Section: Preliminaries Definitionmentioning
“…They have a simple method to describe the model for mechanical systems. Welyczko generalized some of them to the case of 3-dimensional normal almost contact metric manifolds, especially, quasi-Sasakian manifolds . Tripathi et al introduce the concept of ε-almost paracontact manifolds, and in particular, of ε-paraSasakian manifolds .…”
“…Moreover J. Welyczko have studied Legendre curve on quasi Sasakian manifolds  and almost paracontact metric manifolds . Motivated by these works, in this paper we study Legendre curves on 3-dimensional (ε, δ ) trans-Sasakian manifolds.…”
In present paper, we obtain curvature and torsion of Legendre curves in 3-dimensional (ε, δ ) trans-Sasakian manifolds. Also important theorems concerning about biharmonic Legendre curves of (ε, δ ) trans-Sasakian manifolds have been given.
“…• β-para-Sasakian if and only if α = 0 and β = 0 and β is constant, in particular, para-Sasakian if β = −1 , ,…”
Section: N} Such Basis Is Called Aφ-basismentioning
“…Recently, J. We lyczko studied curvature and torsion of Frenet-Legendre curves in 3-dimensional normal almost paracontact metric manifolds . The structures of some classes of 3-dimensional normal almost contact metric manifolds were studied in .…”
Abstract. The aim of present paper is to investigate 3-dimensional ξ-projectively flat andφ-projectively flat normal almost paracontact metric manifolds. As a first step, we proved that if the 3-dimensional normal almost paracontact metric manifold is ξ-projectively flat then ∆β = 0. If additionally β is constant then the manifold is β-para-Sasakian. Later, we proved that a 3-dimensional normal almost paracontact metric manifold isφ-projectively flat if and only if it is an Einstein manifold for α, β =const. Finally, we constructed an example to illustrate the results obtained in previous sections.
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