Ruled and rotational surfaces generated by non-null curves with zero weighted curvature in \((\mathbb{L}^3,ax^2+by^2)\). (English) Zbl 1460.53012
Summary: In this study, firstly we give the weighted curvatures of non-null planar curves in Lorentz-Minkowski space with density \(e^{ax^2+by^2}\) 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 \(e^{ax^2}\), 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 \(e^{ax^2+by^2}\) are studied according to some cases of not all zero constants \(a\) and \(b\). We draw the graphics of obtained surfaces.
MSC:
| 53A35 | Non-Euclidean differential geometry |
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
Keywords:
weighted curvature; Lorentz-Minkowski space; space-like and time-like curves; ruled surface; rotational surfaceReferences:
| [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. · Zbl 1321.53016 |
| [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. · Zbl 1348.53006 |
| [3] | A.T. Ali; Special Smarandache Curves in the Euclidean Space, Int. J. Math. Comb., 2, (2010), 30-36. · Zbl 1215.53005 |
| [4] | A.T. Ali; Position Vectors of curves in the Galilean Space G3, Matematnykn Bechnk, 64, 3 (2012), 200-210. · Zbl 1289.53020 |
| [5] | C. Baikoussis and D.E. Blair; On the Gauss map of ruled surfaces, Glasgow Math. J., 34, (1992), 355-359. · Zbl 0762.53004 |
| [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. · Zbl 1245.53026 |
| [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. · Zbl 0942.53004 |
| [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. · Zbl 0727.53005 |
| [11] | B. Divjak; Curves in Pseudo-Galilean Geometry, Annales Univ. Sci. Budapest., 41, (1998), 117-128. · Zbl 0932.53005 |
| [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. · Zbl 1044.46057 |
| [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. · Zbl 1285.53055 |
| [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. · Zbl 1243.53025 |
| [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. · Zbl 1413.53023 |
| [17] | R. L´opez; Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 1, (2014), 44-107. · Zbl 1312.53022 |
| [18] | F. Morgan; Manifolds with Density, Not. Amer. Math. Soc., 52(8), (2005), 853-858. · Zbl 1118.53022 |
| [19] | F. Morgan; Myers’ Theorem With Density, Kodai Math. J., 29, (2006), 455-461. · Zbl 1132.53306 |
| [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. · Zbl 1370.49041 |
| [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. · Zbl 1431.53006 |
| [22] | A. Turgut and H.H. Hacısalihoˇglu; Timelike Ruled Surfaces in the Minkowski 3-Space-II, Tr. J. of Mathematics, 22, (1998) , 33-46. · Zbl 1006.53018 |
| [23] | M. Turgut and S. Yilmaz; Smarandache Curves in Minkowski Space-time, Int. J. Math. Comb., 3, (2008), 51-55. · Zbl 1202.53006 |
| [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. · Zbl 1423.53006 |
| [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. · Zbl 1380.53019 |
| [26] | D. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
