A kind of rotational surfaces with a light-like axis in conformally flat pseudo-spaces of dimensional three

Year 2024, Volume: 73 Issue: 2, 450 - 459, 21.06.2024

Fırat Yerlikaya

https://doi.org/10.31801/cfsuasmas.1337217

Abstract

In this work, we define the rotational surface with a light-like axis in conformally flat pseudo-spaces $\left(\mathbb{E}_3^1\right)_\lambda$, where $\lambda$ is a radial-type conformal factor. We relate the principal curvatures of a non-degenerate surface that belongs to conformally equivalent spaces $\left(\mathbb{E}_3^1\right)_\lambda$ and $\mathbb{R}_1^3$, based on the radial conformal factor. Thus, we establish a relationship between the radial conformal factor and the profile curve of the rotational flat surface in $\left(\mathbb{E}_3^1\right)_\lambda$, but also for that of the rotational surface with zero extrinsic curvature.

Keywords

Rotation surface, light-like axis, conformally flat pseudo-space, conformally flat pseudo-metric

References

• Wang, G. J., Tang, K.. Tai, C. L., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des., 36(5) (2004), 447-459. https://doi.org/10.1016/S0010-4485(03)00117-9
• Bayram, E., Güler, F., Kasap, E., Parametric representation of a surface pencil with a common asymptotic curve, Comput. Aided Des., 44(7) (2012), 637-643. https://doi.org/10.1016/j.cad.2012.02.007
• Yüzbaşı, Z. K., Yoon, D. W., Characterizations of a helicoidal and a catenoid, Hacet. J. Math. Stat., 51(4) (2022), 1005-1012. https://doi.org/10.15672/hujms.881876
• Arslan, K., Bulca, B., Kosova, D., On generalized rotational surfaces in Euclidean spaces, J. Korean Math., 54(3) (2017), 999-1013. https://doi.org/10.4134/JKMS.j160330
• Altin, M., Kazan, A., Karadag, H. B., Rotational surfaces generated by planar curves in $\mathbb{E}^{3}$ with density, I. J. Analy. Appl., 17(3) (2019), 311-328. https://doi.org/10.28924/2291-8639
• Kazan, A., Altın, M., Yoon, D. W., Generalized rotation surfaces in $\mathbb{R}^{4}$ with density, J. Geom. Phys., 186 (2023), 1-10. https://doi.org/10.1016/j.geomphys.2023.104770
• Corro, A. V., Pina, R., Souza, M. A., Surfaces of rotation with constant extrinsic curvature in a conformally flat 3-space, Results Math., 60 (2011), 225-234. https://doi.org/10.1007/s00025-011-0172-3
• Corro, A. V., Souza, M. A., Pina, R., Classes of Weingarten surfaces in $S^{2}\times R$, Houston J. Math., 46(3) (2020), 651-654.
• Araujo, K., Corro, A., Pina, R., Souza, M., Complete surfaces with zero curvatures in conformally flat spaces, Publ. Math. Debrecen, 96(3-4) (2020), 363-376. https://doi.org/10.5486/PMD.2020.8669
• Lee, C. W., Lee, J. W., Yoon, D. W., Helicoidal surfaces with prescribed curvatures in a conformally flat 3-space, Georgian Math. J., 28(5) (2021), 755-763. https://doi.org/10.1515/gmj-2020-2087
• Lima, B. P., Souza, P. A., Vieira, B. M., Helicoidal hypersurfaces and graphs in conformally flat spaces, Results Math., 77(119) (2022), 1-16. https://doi.org/10.1007/s00025-022-01658-9
• Araujo, K. O., Cui, N., Pina, R. S., Helicoidal minimal surfaces in a conformally flat 3- space, Bull. Korean Math., 53(2) (2016), 531-540. https://doi.org/10.4134/BKMS.2016.53.2.531
• Lee, C. W., Lee, J. W., Yoon, D. W., On helicoidal surfaces in a conformally flat 3-space, Mediterr. J. Math., 164(14) (2017), 1-9. https://doi.org/10.1007/s00009-017-0967-x
• Yerlikaya, F., Helicoidal surfaces in some conformally flat pseudo-spaces of dimensional three, Int. J. Geom. Methods Mod. Phys., 20(3) (2023), 1-23. https://doi.org/10.1142/S0219887823500524
• Sancı, B., Yerlikaya, F., Helicoidal surfaces with prescribed curvatures in some conformally flat pseudo-spaces of dimensional three, Fundam. J. Math. Appl., 6(3) (2023), 157-169. https://doi.org/10.33401/fujma.1315178
• Duggal, K. L., Dae, H. J., Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, 2007.
• Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, 1976.