Zehra Özdemir | Searcharxiv

arXiv2022

2-Ruled hypersurfaces in Minkowski 4-space and their constructions via octonions

In this paper, we define three types of 2-Ruled hypersurfaces in the Minkowski 4-space $\mathbb{E}^4_1$. We obtain Gaussian and mean curvatures of the 2-ruled hypersurfaces of type-1 and type-2, and some characterizations about its minimality. We also deal with the first Laplace-Beltrami operators of these types of 2-Ruled hypersurfaces in $\mathbb{E}^4_1$. Moreover, the importance of this paper is that the definition of these surfaces by using the octonions in $\mathbb{E}^4_1$. Thus, this is a new idea and make the paper original. We give an example of 2-ruled hypersurface constructed by octonion and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one-dimensional object embedded in the 4-dimensional Minkowski space $\mathbb{E}^4_1$. Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2-ruled hypersurfaces in a four-dimensional Minkowski space.

math.DG↗

arXiv2021

Elliptical trajectories of a point on the elliptical 2-sphere

The focus of this work is to analyze the trajectories of a point on the ellipsoid $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$ while it is under the influence of a Killing vector field $K$. For this purpose, we introduce the generalized Darboux frame and the variational vector fields of $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. Then, we determine the Killing equations in terms of the Darboux frame invariants along an ellipsoidal curve. The Killing equations make it possible for us to interpret the magnetic trajectory of a point on the ellipsoid $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. Then, we determine two special trajectories using the variational method. The first one is magnetic curves that are the trajectories produced by the Killing magnetic field $K$ are satisfied the following Lorentz force equation $F_{L} (t)=K\times _{E}t=\nabla _{T}t$, where $\times _{E}$ is elliptical cross product and $\nabla $ is the Levi-Civita connection of the ellipsoid $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. The second one is generalized magnetic helices that are trajectories described by the trajectory of a point on a great ellipse of the ellipsoid rolling without slipping on a fixed ellipse of the ellipsoid using the elliptical motion on the $\mathbb{S}_{a_{1},a_{2},a_{3}}^{2}$. Furthermore, we give various examples and visualized them with the program Mathematica.

math.GM↗