Rotational hypersurfaces in Lorentz-Minkowski 4-space

Year 2021, Volume: 50 Issue: 5, 1409 - 1433, 15.10.2021

Mustafa Altın , Ahmet Kazan

https://doi.org/10.15672/hujms.826596

Cited By: 2

Abstract

In this study, we study rotational hypersurfaces in 4-dimensional Lorentz-Minkowski space. We find the rotational hypersurfaces about spacelike axis according to Gaussian and mean curvatures in $E_{1}^{4}$ and give some results with the aid of the Gaussian and mean curvatures. After that, we deal with the Gauss map of rotational hypersurface about spacelike axis by obtaining the Gaussian and mean curvatures. We obtain the second and third Laplace-Beltrami operators on rotational hypersurface about spacelike axis in $E_{1}^{4}$. Also, we give these characterizations for rotational hypersurfaces about timelike and lightlike axes, too.

Keywords

Rotational hypersurface, Gauss map, Second Laplace-Beltrami operator, Third Laplace-Beltrami operator

References

• [1] M. Altın, A. Kazan and H.B. Karadağ, Monge Hypersurfaces in Euclidean 4-Space with Density, Journal of Polytechnic, 23 (1), 207–214, 2020.
• [2] K. Arslan, B.K. Bayram, B. Bulca and G. Öztürk, Generalized Rotation Surfaces in $E^{4}$, Results. Math. 61, 315–327, 2012.
• [3] K. Arslan, B.K. Bayram, B. Bulca and G. Öztürk, On translation surfaces in 4- dimensional Euclidean space, Acta et Com. Uni. Tar. De Math. 20 (2), 123–133, 2016.
• [4] A. Arvanitoyeorgos, G. Kaimakamis and M. Magid, Lorentz Hypersurfaces in $E_{1}^{4}$ satisfying $\Delta\vec{H}=\alpha\vec{H},$ Illinois J. Math. 53 (2), 581–590, 2009.
• [5] M. Bekkar and B. Senoussi, Translation surfaces in the 3-dimensional space satisfying $\Delta^{III}r_{i}=\mu_{i}r_{i},$ J. Geom. 103, 367–374, 2012.
• [6] B.Y. Chen and S. Ishikawa, On classification of some surfaces of revolution of finite type, Tsukuba J. Math. 17 (1), 287–298, 1993.
• [7] Q-M. Cheng and Q-R. Wan, Complete Hypersurfaces of $R^{4}$ with Constant Mean Curvature, Monatsh. Math. 118, 171–204, 1994.
• [8] A. Çakmak, M.K. Karacan, S. Kızıltuğ and D.W. Yoon, Translation Surfaces in the 3-Dimensional Galilean Space Satisfying $\Delta^{II}x_{i}=\lambda_{i}x_{i}$, Bull. Korean Math. Soc. 54 (4), 1241–1254, 2017.
• [9] U. Dursun, Rotational Hypersurfaces in Lorentz-Minkowski Space with Constant Mean Curvature, Taiwanese J. Math. 14 (2), 685–705, 2010.
• [10] G. Ganchev and V. Milousheva, General rotational surfaces in the 4-dimensional Minkowski space, Turkish J. Math. 38, 883–895, 2014.
• [11] E. Güler, Helical Hypersurfaces in Minkowski Geometry $E_{1}^{4}$ , Symmetry, 12, 1206, 2020.
• [12] E. Güler and A.G. Karaalp, Dini Type Helicoidal Hypersurface in 4-Space, Ikonian Journal of Mathematics 1 (1), 26–34, 2019.
• [13] E. Güler and Ö. Kişi, Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space $E_{1}^{4}$, Mathematics, 7 (205), 1–8, 2019.
• [14] E. Güler, H.H. Hacısalihoğlu and Y.H. Kim, The Gauss map and the third Laplace- Beltrami operator of the rotational hypersurface in 4-Space, Symmetry, 10 (9), 1–11, 2018.
• [15] T.H. Hasanis and T.H. Vlachos, Hypersurfaces in $E^{4}$ with Harmonic Mean Curvature Vector Field, Math. Nachr. 172, 145–169, 1995.
• [16] S. Izumiya, M.D.C.R. Fuster and K. Saji, Flat Lightlike Hypersurfaces in Lorentz- Minkowski 4-Space, J. Geom. Phys. 59, 1528–1546, 2009.
• [17] C. Moore, Surfaces of rotation in a space of four dimensions, Ann. Math. 21, 81–93, 1919.
• [18] M. Moruz and M.I. Munteanu, Minimal translation hypersurfaces in $E^{4}$, J. Math. Anal. Appl. 439 (2), 798–812, 2016.
• [19] D.W. Yoon, Rotation surfaces with finite type Gauss map in $E^{4}$, Indian J. Pure Appl. Math. 32 (12), 1803–1808, 2001.