Ruled and Rotational Surfaces Generated by Non-Null Curves with Zero Weighted Curvature in $(\mathbb{L}^{3},ax^{2}+by^{2})$

Year 2020, Volume: 13 Issue: 2, 11 - 29, 15.10.2020

Mustafa Altın , Ahmet Kazan , Hacı Bayram Karadağ

https://doi.org/10.36890/iejg.599817

Cited By: 1

Abstract

In this study, firstly we give the weighted curvatures of non-null planar curves in Lorentz-Minkowski space with density e^(ax2+by2) and we obtain the planar curves whose weighted curvatures vanish in this space according to the cases of not all zero constants a and b. After giving the Frenet vectors of the non-null planar curves with zero weighted curvature in Lorentz-Minkowski space with density e^(ax2), we create the Smarandache curves of them. With the aid of these curves and their Smarandache curves, we get the ruled surfaces whose base curves are non-null curves with vanishing weighted curvature and ruling curves are Smarandache curves of them. Followingly, we give some characterizations for these ruled surfaces by obtaining the mean and Gaussian curvatures, distribution parameters and striction curves of them. Also, rotational surfaces which are generated by non-null planar curves with zero weighted curvatures in Lorentz-Minkowski space E^3_1 with density e^(ax2+by2) are studied according to some cases of not all zero constants a and b. We draw the graphics of obtained surfaces.

Keywords

Weighted curvature, Lorentz-Minkowski space, spacelike and timelike curves, ruled surface, rotational surface

Supporting Institution

İnönü Üniversitesi BAP

Project Number

FDK-2018-1349

Thanks

We're very much thankfull to İnönü University BAP for supporting our study.

References

• [1] HS. Abdel-Aziz and M.K. Saad; Smarandache Curves Of Some Special Curves in the Galilean 3-Space, Honam Mathematical Journal, 37(2), (2015), 253-264.
• [2] A.L. Albujer and M .Caballero; Geometric Properties of Surfaces with the Same Mean Curvature in R3 and L3, J. Math. Anal. Appl., 445, (2017), 1013-1024.
• [3] A.T. Ali; Special Smarandache Curves in the Euclidean Space, Int. J. Math. Comb., 2, (2010), 30-36.
• [4] A.T. Ali; Position Vectors of curves in the Galilean Space G3, Matematnykn Bechnk, 64, 3 (2012), 200–210.
• [5] C. Baikoussis and D.E. Blair; On the Gauss map of ruled surfaces, Glasgow Math. J., 34, (1992), 355-359.
• [6] L. Belarbi and M. Belkhelfa; Surfaces in R3 with Density, i-manager’s Journal on Mathematics, 1(1), (2012), 34-48.
• [7] J.H. Choi, Y.H. Kim and A.T. Ali; Some associated curves of Frenet non-lightlike curves in E31 ; J. Math. Anal. Appl., 394, (2012), 712–723.
• [8] I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu; Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7(1), (2006), 1-15.
• [9] F. Dillen and W. K¨uhnel; Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math., 98, (1999), 307–320.
• [10] F. Dillen, J. Pas and L. Verstraelen; On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica, 18, (1990), 239-246.
• [11] B. Divjak; Curves in Pseudo-Galilean Geometry, Annales Univ. Sci. Budapest., 41, (1998), 117-128.
• [12] C. Ekici and H. ¨ Ozt¨urk; On Time-Like Ruled Surfaces in Minkowski 3-Space, Universal Journal of Applied Science, 1(2), (2013), 56-63.
• [13] M. Gromov; Isoperimetry of waists and concentration of maps, Geom. Func. Anal., 13, (2003), 178-215.
• [14] D.T. Hieu and T.L. Nam; The classification of constant weighted curvature curves in the plane with a log-linear density, Commun. Pure Appl. Anal., 13, (2013), 1641-1652.
• [15] A. Kazan and H.B. Karada˘g; A Classification of Surfaces of Revolution in Lorentz-Minkowski Space, Int. J. Contemp. Math. Sciences, Vol. 6, no. 39, (2011), 1915-1928.
• [16] A. Kazan and H.B. Karada˘g; Weighted Minimal And Weighted Flat Surfaces of Revolution in Galilean 3-Space with Density, Int. J. Anal. Appl., 16(3), (2018), 414-426.
• [17] R. L´opez; Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 1, (2014), 44–107.
• [18] F. Morgan; Manifolds with Density, Not. Amer. Math. Soc., 52(8), (2005), 853-858.
• [19] F. Morgan; Myers’ Theorem With Density, Kodai Math. J., 29, (2006), 455-461.
• [20] T.L. Nam; Some results on curves in the plane with log-linear density, Asian-European J. of Math., 10(2), (2017), 1-8.
• [21] S. S¸enyurt, Y. Altun and C. Cevahir; Smarandache curves for spherical indicatrix of the Bertrand curves pair, Boletim da Sociedade Paranaense de Matematica, 38(2), (2020), In Press, 27-39.
• [22] A. Turgut and H.H. Hacısalihoˇglu; Timelike Ruled Surfaces in the Minkowski 3-Space-II, Tr. J. of Mathematics, 22, (1998) , 33-46.
• [23] M. Turgut and S. Yilmaz; Smarandache Curves in Minkowski Space-time, Int. J. Math. Comb., 3, (2008), 51-55.
• [24] D.W. Yoon, D-S. Kim, Y.H. Kim and J.W. Lee; Constructions of Helicoidal Surfaces in Euclidean Space with Density, Symmetry, 173, (2017), 1-9.
• [25] D.W. Yoon; Weighted Minimal Translation Surfaces in Minkowski 3-space with Density, International Journal of Geometric Methods in Modern Physics, 14(12), (2017), 1-10.
• [26] D.W. Yoon and Z.K. Y¨uzba¸sı; Weighted Minimal Affine Translation Surfaces in Euclidean Space with Density, International Journal of Geometric Methods in Modern Physics, 15(11), (2018).