Rulings on the Surfaces Having Null Axis and Null Profile Curve in Lorentz-Minkowski 3-Space

Year 2022, Volume: 15 Issue: 1, 11 - 19, 30.04.2022

Erhan Güler

https://doi.org/10.36890/iejg.1064158

Cited By: 1

Abstract

We define the curves family of the surfaces with null profile curve and null axis, and give some smooth functions in three dimensional Lorentz-Minkowski space $\mathbb{L}^{3}$. In addition, we compute the third Laplace-Beltrami operator of this type surfaces.

Keywords

Lorentz-Minkowski space, rulings, surface, null axis, null profile curve, curvature

References

• Baikoussis, Chr., Koufogiorgos, T.: Helicoidal surfaces with prescribed mean or a Gaussian curvature. J. Geom. 63, 25–29 (1998).
• Beneki, Chr. C., Kaimakamis, G., Papantoniou, B.J.: A classification of surfaces of revolution of constant Gaussian curvature in the Minkowski space R3 1. Bull. Calcutta Math. Soc. 90, 441–458 (1998).
• Beneki, Chr. C., Kaimakamis, G., Papantoniou, B.J.: Helicoidal surfaces in three-dimensional Minkowski space. J. Math. Anal. Appl. 275, 586–614 (2002).
• Bobenko, A.I.: Constant mean curvature surfaces and integrable equations. Russian Math. Soc. 46, 1–45 (1991).
• Do Carmo, M.P., Dajczer, M.: Helicoidal surfaces with constant mean curvature. Tohôku Math. J. 34, 425–435 (1982).
• Choi, S.M.: On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space. Tsukuba J. Math. 19, 351–366 (1996).
• Crambin, M., Pirani, F.A.E.: Applicable Differential Geometry. London Math. Soc. Lecture Notes Series, Vol. 59, Cambridge Un. Press, London, (1986).
• Dillen, F., Kühnel, W.: Ruled Weingarten surfaces in Minkowski 3-space. Manuscripta Math. 98, 307–320 (1999).
• Eisenhart, L.: A Treastise on the Differential Geometry of Curves and Surfaces. Palermo 41 Ginn and Company, USA (1909).
• Güler, E.: Bour’s theorem and light-like profile curve. Yokohama Math. J. 54 (1) 55–77 (2007).
• Güler, E., Vanlı, A.: Bour’s theorem in Minkowski 3-space. J. Math. Kyoto. 46 (1) 47–63 (2006).
• Güler, E., Yaylı, Y., Hacısaliho˘glu, H.H.: Bour’s theorem on Gauss map in Euclidean 3-space. Hacett. J. Math. Stat., 39 (4), 515–525 (2010).
• Hano, J., Nomizu, K.: Surfaces of revolution with constant mean curvature in Lorentz-Minkowski space. Tohoku Math. J. 36, 427–437 (1984).
• Hitt, L., Roussos, I.: Computer graphics of helicoidal surfaces with constant mean curvature. An. Acad. Brasil. Ciênc. 63, 211–228 (1991).
• Kaimakamis, G., Papantoniou, B., Petoumenos, K.: Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying ∆III r = Ar. Bull. Greek Math. Soc. 50, 75–90 (2005).
• Kenmotsu, K.: Surfaces of revolution with prescribed mean curvature. Tohôku Math. J. 32 (1980), 147–153.
• Sasahara, N.: Spacelike helicoidal surfaces with constant mean curvature in Minkowski 3-space. Tokyo J.Math. 23, 477–502 (2000).
• Struik, D.J.: Lectures on Differential Geometry. Addison-Wesley, New-York, (1961).