Ruled Surfaces and Tangent Bundle of Pseudo-Sphere of Natural Lift Curves

Abstract

In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, TM, and Lorentzian unit sphere, S12 is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, TM, and hyperbolic unit sphere, H2 is given. According to E. Study mapping, any curve on S12 or H2 corresponds to a ruled surface in R13. By constructing these isomorphisms, we correspond to any natural lift curve on TM or TM a unique ruled surface in R13. 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.

Intersections of Two Ruled Surfaces Corresponding to Curves on the Unit Dual Sphere

“…Theorem 2. Let be the curve on the unit dual sphere In the ruled surface generated by the curve on the unit dual sphere is shown as (4) where is the base curve of the ruled surface see [12,13].…”

“…Moreover, it has several applications in engineering, architecture, physics, surface modelling, etc. Especially, the ruled surface, which is obtained by moving a line along the curve, has many applications and its geometric interpretation has been studied by a lot of authors such as [3][4][5][6][7]. However, there are many researches about the theory of surfaces while there are a little researches about the intersection of surfaces in literature.…”

Intersections of Ruled Surfaces Corresponding to Curves on Pseudo Spheres in Dual Space

“…). In the ruled surface generated by the curve on the dual Lorentz unit sphere (or the dual hyperbolic unit sphere) is shown as (5) where is the base curve of the ruled surface see [16].…”

“…spacelike) lines in and ordered pairs of vectors on and see for detail in [15]. Utilizing E. Study mapping, the curve on the dual Lorentz unit sphere (or dual hyperbolic unit sphere) corresponds to the ruled surfaces in Lorentzian space, see [16] Additionally, some geometric properties of ruled surfaces produced by natural lift curves in are investigated in the same study. Also, for detailed information about surfaces in Lorentz space and its properties, see [17].…”

On Ruled Surfaces Constructed by the Evolution of a Polynomial Space Curve

“…To solve this problem, Dede defined the Flc frame for moving polynomial curves [7,8]. The Flc frame [9][10][11][12][13] and ruled surfaces on different frames [14][15][16][17][18][19][20][21][22][23] have been investigated by many researchers. Inspired by these studies, we conducted this research to create a new resource on the subject of surfaces and to form a basis for future studies.…”