Evolution Equations of Pseudo Spherical Images for Timelike Curves in Minkowski 3-Space

Abstract

The pseudo spherical images of non-lightlike curves in Minkowski geometry are curves on the unit pseudo sphere, which are intimately related to the curvatures of the original ones. These images are obtained by means of Frenet-Serret frame vector fields associated with the curves. This classical topic is a well-known concept in Lorentzian geometry of curves. In this paper, we introduce the pseudo spherical images for a timelike curve in Minkowski 3-space. Our main purpose of the work is to obtain the time evolution equations of the orthonormal frame and curvatures of these images. The compatibility conditions for the evolutions are used. Finally, the theoretical results obtained through this study are given by some important theorems and explained in two computational examples with the corresponding graphs.

Investigation of Curvature Functions of Special Ruled Surfaces in Euclidean 3-Space

“…In future works, we plan to study these surfaces in Lorentz-Minkowski space for different queries and further improve the results in this paper, combined with the techniques and results in [24].…”

Spacelike curves and evolution equations in de Sitter 3-space

“…In future works, we plan to study the geometry of moving spacelike and timelike curves in different spaces like Galilean and pseudo-Galilean spaces for different queries and further improve the results in this paper, combined with the techniques and results in [18][19][20][21].…”

“…Theorem 3.3 Let r(s, t) be an inelastic spacelike curve in S 3 1 and consider Frenet matrix Ω that satisfies Eqs. (10) and (21). Then, we have the integrability condition:…”

Normal Surfaces along a Curve on a Surface in Euclidean 3-Space

“…Moreover, we have illustrated the convenience and efficiency of this approach by some representative examples. In future works, we plan to study the normal surfaces in Lorentz-Minkowski space for different queries and further improve the results in this paper, combined with the techniques and results in the related papers [11][12][13][14][15][16].…”