Some Characterizations of Translation Surface Generated by Spherical Indicatrices of Timelike Curves in Minkowski 3-space

In this paper, we study translation surfaces generated by spherical indicatrices of timelike curves in Minkowski 3-space and find necessary and sufficient conditions for the translation surfaces to be flat or minimal. Further, we obtain necessary and sufficient conditions for generating curves of the translation surfaces to be geodesic, asymptotic line and line of curvature. Finally for such translation surfaces we obtain the axis when they are constant angle surfaces.

Keywords:

Tangent indicatrices, principal normal indicatrices binormal indicatrices, timelike curves,

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[1] Ali, A. T.: Non-lightlike constant angle ruled surfaces in Minkowski 3-space. Journal of Geometry and Physics. 157 (1-2), 103833 (2020). https://doi.org/10.1016/j.geomphys.2020.103833
[2] Ali, A. T., Abdel-Aziz, H. S., Sorour, A. H.: On curvatures and points of the translation surfaces in Euclidean 3-space. J. Egyptian Math. Soc. 23 (1), 167-172 (2015). https://doi.org/10.1016/j.joems.2014.02.007
[3] Arfah, A.: On Causal Characterization of Spherical Indicatrices of Timelike Curves in Minkowski 3-Space. Hagia Safia J. of Geometry. 2 (2), 26-37 (2020).
[4] Acar, N., Aksoyak, F. K.: Some Characterizations of translation surfaces generated by spherical indicatrices of space curves. Palestine Journal of Mathematics. 11 (1), 456-468 (2022).
[5] Baba-Hamed, Ch., Mohammed, B., Zoubir, H.: Translation Surfaces in the Three-Dimensional Lorentz-Minkowski Space Satisfying $\Delta r_i = \lambda_i r_i$. Int. J. Math. Anal. 4 (17), 797-808 (2010).
[6] Barros, M.: General helices and a theorem of Lancret. Proc. American Mathematical Soc. 125 (5), 1503-1509 (1997).
[7] Çetin, M., Kocayiğit, H., Önder, M.: Translation surfaces according to Frenet frame in Minkowski 3-space. International Journal of Physical Sciences. 7 (47), 6135-6143 (2012).
[8] Çetin, M., Tunçer, Y.: Parallel surfaces to translation surfaces in Euclidean 3- space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 64 (2), 47-54 (2015).
[9] Couto, I. T., Lymberopoulas, A.: Introduction to Lorentz Geometry, curves and surfaces. CRC Press. Florida (2021). https://doi.org/10.1201/9781003031574-1
[10] do Carmo, M. P.: Differential geometry of curves and surfaces. Prentice-Hall Inc. New Jersey (1976).
[11] Fu, Y., Yang, D.: On constant slope Spacelike surfaces in 3-dimensional Minkowski space. Journal of Mathematical Analysis and Application. 385 (1), 208-220 (2012). https://doi.org/10.1016/j.jmaa.2011.06.040
[12] Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk. J. Math. 28 (2), 531-537 (2004).
[13] Kılıç, O., Çalışkan, A.: The Frenet and Darboux Instantaneous Rotation Vectors for curves and spacelike Surfaces. Mathematical and Computational Applications. 1 (2), 77-86 (1996).
[14] Kühnel, W.: Differential Geometry: Curves - Surfaces - Manifolds. American Math. Soc. Rhode Island (2015).
[15] Liu, H.: Translation surfaces with constant mean curvature in 3-dimensional spaces. Journal of Geometry. 64 (1-2), 141-149 (1999). https://doi.org/10.1007/BF01229219
[16] López, R., Munteanu, M. I.: Constant angle surfaces in Minkowski space. Bulletin of the Belgian Math. Soc. 18 (2), 271-286 (2011). https://doi.org/10.36045/bbms/1307452077
[17] Munteanu, M. I., Nistor, A. I.: On the geometry of the second fundamental form of translation surfaces in $E^3$ . Houston J. Math. 37 (4), 1087-1102 (2011).
[18] O’Neill, B.: Semi-Riemannian geometry with application to relativity. Academic Press. London (1983).
[19] Pressley, A.: Elementary differential geometry. Springer. London (2001).
[20] Ratcliffe, J. G.: Foundations of Hyperbolic Manifolds. Springer. New York (2006).
[21] Ünlütürk, Y., Savcı, Ü. Z.: On non-unit speed curves in Mikowski 3-space. Scientia Magna. 8 (4), 66-74 (2012).
[22] Yılmaz, S., Özyılmaz, E., Turgut, M.: New Spherical Indicatrices and their Characterizations. Analele stiintifice ale Universitatii Ovidius Constanta. 18 (2), 337-354 (2010).