In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, T (M) over tilde, and Lorentzian unit sphere, S-1(2) is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, T (M) over cap, and hyperbolic unit sphere, H-2 is given. According to E. Study mapping, any curve on S-1(2) or H-2 corresponds to a ruled surface in R-1(3). By constructing these isomorphisms, we correspond to any natural lift curve on T (M) over tilde or T (M) over cap a unique ruled surface in R-1(3). Then we calculate striction curve, shape operator, Gaussian curvature and mean curvature of these ruled surfaces. We give developability condition of these ruled surfaces. Finally, we give examples to support the main results.

It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the $4\times 4$ matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

In this paper, an isomorphism between unit dual sphere, DS2, and the subset of tangent bundle of unit 2-sphere, T￣M, is represented. According to E. Study mapping, a ruled surface in IR3 corresponds to each curve on DS2. Through this isomorphism, new forms of ruled surfaces called slant ruled surfaces in IR3 were introduced. Moreover, conditions for these surfaces to be slant ruled surfaces were given. Finally, a unique ￣q−,￣h− and ā− slant ruled surfaces in IR3 were corresponded to each striction curve of natural lift curve on T￣M.

This paper aims to present, in a unified manner, results which are valid on both split quaternions with quaternion coefficients and quaternions with dual coefficients, simultaneously, calling the attention to the main differences between these two quaternions. Taking into account some results obtained by Karaca, E. et al., 2020, each of these quaternions is studied and some important differences are remarked on.

The purpose of this paper is to define and construct new number systems, called the harmonic complex Fibonacci sequences (HCF) and the harmonic hybrid Fibonacci (HHF) sequences. These sequences are defined by inspiring the well-known harmonic and hybrid numbers in literature. We give some fundamental definitions and theorems about these sequences in detail. Moreover, we examine some algebraic properties such as Binet-like-formula, partial sums related to these sequences. Finally, we provide a Maple 13 source code to verify the sequences easily.