Ruled Surfaces and Tangent Bundle of Unit 2-Sphere of Natural Lift Curves

Abstract

This article deals with the isomorphism between unit dual sphere, DS^2, and the subset of the tangent bundle of unit 2-sphere, TM ̅. According to E. Study mapping, a ruled surface in〖 R〗^3corresponds to each curve on DS^2. Through this correspondence, a unique ruled surface in R^3is corresponded to natural lift curve on TM ̅. Then striction curve, shape operator, mean curvature and Gaussian curvature of these ruled surfaces obtained by the natural lift curves are calculated. Developabilitiy condition of these ruled surfaces is given. Finally, we give an example to support the main results.

Keywords

• Natural lift,Tangent bundle,Unit dual sphere,Ruled surface,Study’s map

References

1. Referans 1 Hathout, F., Bekar, M., Yaylı, Y., “Ruled surfaces and tangent bundle of unit 2-sphere” International Journal of Geometric Methods in Modern Physics, 2: (2017).
2. Referans 2 Bekar, M., Hathout, F., Yaylı, Y., “Tangent bundle of pseudo-sphere and ruled surfaces in Minkowski 3-space”, General Letters in Mathematics, 5: 58-70, (2018).
3. Referans 3 Ergün, E., Çalışkan, M., “On natural lift of a curve”, Pure Mathematical Sciences, 2: 81-85, (2012).