Mustafa CALISKAN, Emel KARACA

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

This article deals with the isomorphism between unit dual sphere, ??2, and the subset of thetangent bundle of unit 2-sphere, ?? ̅ . According to E. Study mapping, a ruled surfacein ℝ3corresponds to each curve on ??2. Through this correspondence, a unique ruled surface inℝ3is corresponded to natural lift curve on ?? ̅. Then striction curve, shape operator, meancurvature and Gaussian curvature of these ruled surfaces obtained by the natural lift curves arecalculated. Developabilitiy condition of these ruled surfaces is given. Finally, we give an exampleto support the main results.

PDF

___

Thorpe, J.A., “Elementary topics in differential geometry”, Springer Verlag, New York, Heidelberg- Berlin, (1979).
Ergün, E., Çalışkan, M., “On natural lift of a curve”, Pure Mathematical Sciences, 2: 81-85, (2012).
Ergün, E., Bilici, M., Çalışkan, M., “The Frenet vector fields and the curvatures of the natural lift curve”, The Bulletin Society for Mathematical Services and Standarts, 2: 38-43, (2012).
Do Carmo, M., “Differential geometry of curves and surfaces”, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, (1976).
Yaylı, Y., Saraçoğlu, S., “Different approaches to ruled surfaces”, SDU Journal of Science, 7: 56- 68, (2012).
Şentürk, G.Y., Yüce, S., “Characteristic properties of the ruled surface with Darboux frame in ℝ3”, Kuwait J Sci., 2: 14-33, (2015).
Orbay, K., Aydemir, İ., “The ruled surfaces generated by Frenet vectors of a curve in ℝ13”, Cbu Journal of Science, 6: 155-160, (2010).
Izumiya, S., Takeuchi, N., “Special curves and ruled surfaces”, Contributions to Algebra and Geometry, 44: 203-212, (2003).
Şenyurt, S., Çalışkan, A., “The quaternionic expression of ruled surfaces”, Filomat, 32: 5753-5766, (2018).
Hacısalihoğlu, H.H., “On the pitch of a closed ruled surface”, Mechanism and Machine Theory, 7: 291-305, (1972).
Masal, M., Azak, A. Z., “The ruled surfaces according to type-2 Bishop frame in Euclidean 3-Space ?3”, Mathematical Sciences and Applications E-Notes, 3: 74-83, (2015).
Sarıoğlugil, A., Tutar, A., “On ruled surfaces in Euclidean space ?3”, Int. J. Contemp. Math. Sci.,2: 1-11, (2007).
Fischer, I. S., “Dual-Number Methods in Kinematics and Dynamics”, CRC Press, Boca Raton, London, New York, Washington DC: (1999).
Karakaş, B., Gündoğan, H., “A relation among ??2, ??2 and non-cylindirical ruled surfaces”, Mathematical Communications, 8: 9-14, (2003).
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).
Bekar, M., Hathout, F., Yaylı, Y., “Tangent bundle of pseudo-sphere and ruled surfaces in Minkowski space”, General Letters in Mathematics, 5: 58-70, (2018).