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Almost contact curves in trans-Sasakian 3-manifolds

This paper is devoted to the study of curvature and torsion of almost contact curves in trans-Sasakian 3-Manifolds. The conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained.


Introduction
Almost contact curves play a important role in geometry and topology of contact metric manifolds, e.g.

a diffeomorphism of a contact manifold is a contact transformation if and only if any almost contact
curves in a domain of it go to almost contact curves [5]. Several authors have studied almost contact curves in contact geometry such as [1,5,6,9,11,12,16,18]. In [5] Baikoussis and Blair have studied almost contact curves in contact metric 3-manifold and gave the Frenet 3-frame in this class of manifold.
Belkhelfa et al. have extended some of the results of [5] from the Riemannian to the Lorentzian case, and classified all biharmonic almost contac curves in Sasaki-Heisenberg spaces [18]. A result of Blair [see theorem 8.2, p.134 [1]] had been generalized by Welyczko [11] to the case of 3-dimensional Quasi-Sasakian manifolds. Moreover the author had also obtained some interesting properties of non-Frenet almost contact curves in normal almost paracontact metric 3-manifolds [12].Özgür and Tripathi established necessary and sufficient conditions for almost contact curves having parallel mean curvature vector, proper mean curvature vector, being harmonic and being of type AW (k), k = 1, 2, 3; in α-Sasakian manifolds [6]. In [9] Lee characterized almost contact curves in a Sasakian manifold having the following properties: (i) a pseudo-Hermitian parallel mean curvature vector field (ii) a pseudo-Hermitian proper mean curvature vector field in the normal bundle. Recently, Inoguchi and Lee have studied almost contact curves in normal almost contact metric 3-manifold satisfying ∇H = λH or ∇ ⊥ H = λH and gave natural equations for planar biminimal curves [16].
The purpose of this paper is to investigate the properties of almost contact curves in trans-Sasakian 3-manifolds. This paper is organized as follows: In §2 we recall some basic definitions and facts about almost contact metric (in brief a.c.m.) manifolds, trans-Sasakian manifolds and Frenet curves. The curvature, torsion of almost contact curves and the conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained, and finally we construct the examples in §3.

Contact metric manifolds
A (2n+1)-dimensional differentiable manifold M is said to be an almost contact manifold if its structural group GL 2n+1 R of linear frame bundle is reducible to U(n) × {1} (Gray [10]). This is equivalent to existence of a tensor field of type (1, 1), a vector field ξ and a 1-form η satisfying From these conditions one can easily obtain Moreover, scince U(n) × {1} ⊂ SO(2n + 1), M admits a Riemannian metric g satisfying for all X, Y ∈ Γ(T M ). Such a metric is called an associated metric (Sasaki [19]) of the almost contact manifold M. With respect to g, η is metrically dual to ξ, that is A structure (φ, ξ, η, g) on M is called an almost contact metric structure and a manifold M equipped with an almost contact metric structure is said to be an almost contact metric manifold. The fundamental 2-form Φ of the manifold is defined by for all X, Y ∈ Γ(T M ). An almost contact metric manifold M is said to be a contact metric manifold if Φ = dη. Here the exterior derivative dη is defined by On a contact metric manifold, η is contact form, i.e., η ∧ (dη) n = 0 everywhere on M . In particular, η ∧ (dη) n = 0 is a volume element on M so that a contact manifold is orientable. Define a (1, 1) type tensor field h and l by h = 1 2 L ξ φ, lX = R(., ξ)ξ , where L denotes the Lie differentiation and R the curvature tensor respectively. The operators h and l are self-adjoint and satisfy: hξ = lξ = 0 and hφ = −φh. Also we have T r.h = T r.φh = 0. Moreover, if ∇ denotes the Levi-Civita connection on M , then following formulas holds on a contact metric manifold.
On the direct product manifold M × R of an almost contact metric manifold M and the real line R, any tangent vector field can be represented as the form X, f d dt , where X ∈ Γ(T M ) and f is a function on M × R and t is the cartesian coordinate on the real line R.
Define an almost complex structure J on M × R by For an arbitrary almost contact metric 3-manifold M , we have ( [20]): where ∇ is the Levi-Civita connection on M .

trans-Sasakian manifolds
This class of manifolds arose in a natural way from the classification of almost contact metric structures and they appear as a natural generalization of both Sasakian and Kenmotsu manifolds. In [4] Gray Harvella classification of almost Hermite manifolds appear as a class W 4 of Hermitian manifolds which are closely related to locally conformally Kähler manifolds. An almost contact metric structure on a manifold M × R belongs to the class W 4 . The class C 6 ⊕ C 5 [13] coincides with the class of trans-Sasakian structure of type (α, β).

Frenet Curves
Let (M, g) be a Riemannian n−manifold with Levi-Civita connection ∇. A unit speed curve γ : I → M is said to be an r−Frenet curve [1] if there exists an orthonormal r−frame field (E 1 = γ ′ , E 2 , ..., E r ) along γ such that there exist positive smooth functions k 1 , k 2 , ..., k r−1 satisfying The function k r is called the r−th curvature of γ. A Frenet curve is said to be • a geodesic if r = 1, i.e., ∇ γ ′ γ ′ = 0.
• a Riemannian circle if r = 2, and k 1 is non-zero constant.
In case n = 3, we denote by (E 1 , E 2 , E 3 ) = (T, N, B). Then we have the Serret-Frenet equation: The first curvature k = k 1 and the second curvature τ = k 2 are called the geodesic curvature and geodesic torsion of γ, respectively. The vector field N and B are called the unit normal vector field and binormal vector field of γ, respectively.

Almost contact Curves
Let γ : I → M be a curve parameterized by arc-length (the natural parametrization) in an almost contact metric 3-manifold M with Frenet frame (T, N, B).

Definition 3.1 A Frenet curve γ in an almost contact 3-manifold M is said to be an almost contact curve
if it is an integral curve of the contact distribution D = kerη, equivalently, η(γ ′ ) = 0.
We begin with a proposition that will motivate the main result:

p is non -zero constant on I and q is certain function on I.
Proof: Suppose that γ ′ is not collinear with ξ and describe curvature (κ) and torsion (τ ) of γ on I. We is curvature of γ.
Let us suppose that M = R 2 × R + , ω : M → R + and (x, y, z) be cartesian coordinates in M , we define a trans-Sasakian structure on M by Certain almost contact curves in the above class of manifolds are given below: