Bahar Doğan Yazıcı

In this study, our aim is to strengthen the results of the existing literature on the concept of Mannheim curves by using the natural and idiosyncratic structure of Myller configuration. For this purpose, we examine Mannheim curves with Frenet-type frame in Myller configuration for Euclidean 3-space E3 and Euclidean 4-space E4. Since the classical Frenet frames in E3 and E4 are special cases of Frenet-type frames with Myller configuration in E3 and E4, and because of the special and natural structure of the Myller configuration, the Mannheim curves with Frenet frames in E3 and E4 are also special cases of the Mannheim curves with Frenet-type frames in E3 and E4. We derive several relations among the Frenet-type frame elements of Mannheim curves with Myller configuration. Subsequently, numerical examples illustrating the presented concepts are constructed, together with corresponding figures. We also introduce a new perspective that yields a fairly large family of curves for Mannheim curves with Myller configuration in E3. Indeed, this family contains several distinct special cases, and we exemplify only one of them by focusing on the most famous case, the Frenet frame. The main reason for this is that this study opens a new perspective for Mannheim curves, since the values ai(s) are smooth functions and the parameter s ∈ I is the arc-length parameter, where i ∈ {1, 2, 3} (and i ∈ {1, 2, 3, 4} for E4). Moreover, the condition ∑3i=1 ai2(s) = 1 (and ∑4i=1 ai2(s) = 1 for E4) defining these special functions arises from the Myller configuration in E3 (and E4). Consequently, we present several graphs illustrating the relationships between special cases and the general setting studied in this paper and in the existing literature.

In this paper, we introduce a linearly polarized light wave in an optical fiber and rotation of the polarization plane through the Frenet‐type frame with Myller configuration. Since the geometric evaluation and interpretations of a polarized light wave are associated with geometric phase, a new type of geometric phase model has been constructed with Myller configuration. Also, the rotation of the polarization plane is determined by the Fermi–Walker parallel transportation law. Then, this is examined with the Rytov parallel transportation along with the direction of the state of the polarization plane in an optical fiber with Myller configuration. Moreover, the electromagnetic curves obtained by the electric field along the polarization plane of a light wave traveling in an optical fiber are given. Furthermore, we examine the Lorentz force equations and scrutinize the electromagnetic trajectories constructed by the electric field of the light wave traveling in the optical fiber with Myller configuration. Then, we give a brief introduction to the geometric phase model for a curve with a Frenet‐type frame with Myller configuration in a 3D Lie group. Additionally, we give some physical interpretations based on the direction of the electromagnetic curves thanks to the unique and natural structure of Myller configuration in both Euclidean 3‐space and 3D Lie group.

In this work, we define parallel $\mathcal{P}$-equidistant ruled surfaces with the Myller configuration. We investigate the characterization of the striction curves and $\mathcal{P}$-distance of parallel $\mathcal{P}$-equidistant ruled surfaces. In addition, we obtain the relationships between the versor fields of the base curves of parallel $\mathcal{P}$-equidistant ruled surfaces and the relationships between curvatures and invariants with the Myller configuration. Finally, we describe the integral invariants drall, apex angle and pitch for each ruled surface drawn by versor fields.

In this study, we aim to bring together the spinors, a useful concept from mathematics to physics, and Bertrand and Mannheim curves of framed curves in 3D Lie groups, that are a special singular curve in 3D Lie groups. For this purpose, we investigate the spinor representations of Bertrand and Mannheim curves of framed curves in 3D Lie groups with a bi-invariant metric. Based on the natural structures of Bertrand and Mannheim curves of framed curves and Lie groups, this study is a comprehensive study that includes spinor representations of both regular and singular Bertrand and Mannheim curves in 3D Lie groups. It is a comprehensive generalization of all the studies done in the spinor representation of relevant concepts, as it includes studies on spinor representations of both regular and singular Bertrand and Mannheim curves in the literature. Then, we determine the special cases of spinor equations of these special curve couples of framed curves. Additionally, we construct some geometric interpretations and results with respect to them.

In this study, we introduce the generalized Tribonacci hyperbolic spinors and properties of this new special numbers system by the generalized Tribonacci numbers, which are one of the most general form of the third-order recurrence sequences, generalized Tribonacci quaternions, and hyperbolic spinors, which have quite an importance and framework from mathematics to physics. This study especially improves the relations between the hyperbolic spinors and generalized Tribonacci numbers with the help of the generalized Tribonacci split quaternions. Furthermore, we examine some special cases of them and construct both new equalities and fundamental properties such as recurrence relation, Binet formula, generating function, exponential generating function, Poisson generating function, summation formulas, special determinant properties, matrix formula, and special determinant equations. Also, we give some numerical algorithms with respect to the obtained materials. In addition to these, we give a brief introduction for further research: generalized Tribonacci polynomial hyperbolic spinor sequence.

In this study, we determine a new type comprehensive frame, which is called the generalized Frenet‐type frame in three‐dimensional Lie groups with Myller configurations, and it includes several special and classical type frames for Euclidean 3‐space and three‐dimensional Lie groups. After constructing this new comprehensive frame, we obtain derivative formulas with the help of the Lie curvature. In addition, we define some special type curves. The geometry of versor fields along a curve with Frenet‐type frame in three‐dimensional Lie groups with Myller configurations is a generalization of the usual theory of curves. Since this particular relationship, the osculating‐type and rectifying‐type curves with Frenet‐type frame in three‐dimensional Lie groups with Myller configurations include some special cases for osculating and rectifying curves in different spaces.