Some Characterizations of Translation Surface Generated by Spherical Indicatrices of Timelike Curves in Minkowski 3-space
Yıl 2023,
, 48 - 61, 30.04.2023
Akhılesh Yadav
,
Ajay Kumar Yadav
Öz
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.
Destekleyen Kurum
COUNCIL OF SCIENTIFIC & INDUSTRIAL RESEARCH HUMAN RESOURCE DEVELOPMENT GROUP
Proje Numarası
09/013(0953)/2020-EMR-I
Teşekkür
I, Ajay Kumar yadav, a research scholar of Department of Mathematics, Banaras Hindu University, India, submitting a research paper(joint work with Dr. Akhilesh Yadav, Assistant Professor, BHU) entitled , "Some characterizations of translation surface generated by spherical indicatrices of timelike curves in Minkowski 3-space" for possible publication in your esteemed journal. kindly acknowledge the receipt of the manuscript.
Thanking You With Regards,
Kaynakça
- [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).
Yıl 2023,
, 48 - 61, 30.04.2023
Akhılesh Yadav
,
Ajay Kumar Yadav
Proje Numarası
09/013(0953)/2020-EMR-I
Kaynakça
- [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).