On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some examples of such curves. 1. Introduction In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the problem of characterization of a regular curve. In the solution of the problem, the curvature functions and of a regular curve have an effective role. It is known that we can determine the shape and size of a regular curve by using its curvatures and . Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. If the position vector of a regular curve is composed by the Frenet frame vectors of another regular curve , then the curve is called a Smarandache curve [2]. Special Smarandache curves in Euclidean and Minkowski spaces are studied by some authors [3–8]. The curves lying on a pseudohyperbolic space in Minkowski 3-space are characterized in [9]. In this paper, we define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame's vectors of special pseudohyperbolical Smarandache curves. In particular, we prove that special -pseudohyperbolical Smarandache curves do not exist. Besides, we give some examples of special pseudohyperbolical Smarandache curves in Minkowski 3-space. 2. Basic Concepts The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by where is a rectangular Cartesian coordinate system of . Since is an indefinite metric, recall that a nonzero vector can have one of the three Lorentzian causal characters: it can be spacelike if , timelike if , and null (lightlike) if . In particular, the norm (length) of a vector is given by and two vectors and are said to be orthogonal if . Next, recall that an arbitrary curve in can locally be spacelike, timelike, or null (lightlike) if all of its