Null Quaternionic Curves in Semi-Euclidean 3-Space of Index ν

Abstract

The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843. The space of quaternions Q are isomorphic to E, 4dimensional vector space over the real numbers. The set of the quaternions, which was introduced by Hamilton, can be represented as Q = {ae1 + be2 + ce3 + d; a, b, c, d ∈ R}. Here, ei = −1, ei × ej = ek, 1 ≤ i ≤ 3 and (ijk) is an even permutation of (123). There are a lot of papers associated with quaternions. Also they are used in both theoretical and applied mathematics. Hence, the importance of the study of quaternions and its presence in the physical theories are clear. In 1987, the Serret-Frenet formulae for quaternionic curves in E and E were given by Bharathi and Nagaraj [1]. After that, split quaternions are identified with semi-Euclidean space E 2 , while the vector part of split quaternions are identified with Minkowski 3-space by Inoguchi [2]. Many studies have been published on the quaternionic curves using this results. Among them Tuna has defined the Serret–Frenet formulae for a quaternionic curve in the semi-Euclidean space E 2 [3]. We have studied quaternion valued functions and quaternionic inclined curves in the semi-Euclidean space E 2 [4]. But, to our knowledge, there has been no study on the Serret– Frenet formulae for null quaternionic curves in the semiEuclidean spaces. In this paper, we will study Serret– Frenet formulae for null quaternionic curve in R1 and null quaternionic curve in R2. It is well known that there exist spacelike quaternionic curve and timelike quaternionic curve in the semiEuclidean spaces. However, null quaternionic curves have many properties which are very different from spacelike quaternionic and timelike quaternionic curves. In geometry of null quaternionic curves difficulties arise because the arc length vanishes, so that it is impossible to normalize the tangent vector in the usual way. Since the length