Klzlltuǧ, Sezai; Erişir, Tülay; Mumcu, Gökhan

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields = 1 + 2n, where 2 1 + 2 2 = 1 for the spatial quaternionic curve and = 1 + 2n + 3β2, where 21 + 22 + 23 = 1 for the quaternionic curve . Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples.

Erişir, Tülay; Güngör, Mehmet Ali

Spinors are used in physics quite extensively. Basically, the forms of use include Dirac four-spinors, Pauli three-spinors and quaternions. Quaternions in mathematics are essentially equivalent to Pauli spin matrices which can be generated by regarding a quaternion matrix as compound. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Fibonacci spinors using the Fibonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini formulas which are given for some series of numbers in mathematics for Fibonacci spinors.

Altunbas, Murat; Bilen, Lokman; Gezer, Aydin

The paper is concerned with the Kaluza-Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.