Rotational surfaces with Cheng-Yau operator in Galilean 3-spaces

Abstract

In this paper, we study three types of rotational surfaces in Galilean 3-spaces. We classify rotational surfaces satisfying $$L_1G=F(G+C)$$ for some constant vector $C\in \mathbb{G}^3$ and smooth function $F$, where $L_1$ denotes the Cheng-Yau operator.

Keywords

• Cheng-Yau operator
• Rotational surfaces
• Galilean spaces
• $L_1$-pointwise 1-type

References

1. [1] L.J. Alias and N. Gurbuz, An extension of Takashi theorem for the linearized operators of the highest order mean curvatures, Geom. Dedicata 121, 113-127, 2006.
2. [2] M.E. Aydin, A.O. Oğrenmis and M. Ergut, Classification of factorable surfaces in the pseudo-Galilean space, Glas. Mat. Ser. III 50 (2), 441-451, 2015.
3. [3] B.Y. Chen, Total mean curvature and submanifolds finite type, World Scientific Publ., New Jersey, 1984.
4. [4] B.Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22, 117-337, 1996.
5. [5] B.Y. Chen, M. Choi, and Y.H. Kim, Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. Soc. 42, 447-455, 2005.
6. [6] B.Y. Chen, J. Morvan and T. Nore, Energy, tension and finite type maps, Kodai Math. J. 9, 406-418, 1986.
7. [7] B.Y. Chen, and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (2), 161-186, 1987.
8. [8] S.Y. Cheng and S.T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225, 195-204, 1977.

9. [9] M. Dede, Tubular surfaces in Galilean space, Math. Commun. 18, 209-217, 2013.
10. [10] M. Dede, C. Ekici and W. Goemans, Surfaces of revolution with vanishing curvature in Galilean 3-space, J. Math. Physics, Analysis, Geometry 14 (2), 141-152, 2018.
11. [11] U. Dursun, Hypersurfaces with pointwise 1-type Gauss map, Taiwanese J. Math. 11 (5), 1407-1416, 2007.
12. [12] S.M.B. Kashani, On some $L_1-$finite type (hyper)surfaces in $\mathbb R^{n+1}$, Bull. Korean Math. Soc. 46 (1), 35-43, 2009.
13. [13] U.H. Ki, D.S. Kim, Y. H. Kim and Y.M. Roh, Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space, Taiwan. J. Math. 13 (1), 317-338, 2009.
14. [14] D.S. Kim, J.R. Kim and Y. H. Kim, Cheng-Yau operator and Gauss map of surfaces of revolution, Bull. Malays. Math. Sci. Soc. 39 (4), 1319-1327, 2016.
15. [15] Y.H. Kim and N.C. Turgay, Surfaces in $\mathbb{E}^3$ with $L_1-$pointwise 1-type Gauss map, Bull. Korean Math. Soc. 50 (3), 935-949, 2013.
16. [16] Y.H. Kim and N.C. Turgay, On the helicoidal surfaces in $\mathbb{E}^3$ with $L_1-$pointwise 1-type Gauss map, Bull. Korean Math. Soc. 50, 1345-1356, 2013.
17. [17] Y.H. Kim and D.W. Yoon, Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (3-4), 191-205, 2000.
18. [18] Z.M. Sipus, Ruled Weingarten surfaces in Galilean space, Period. Math. Hungar. 56, 213-225, 2008.
19. [19] Z.M. Sipus and B. Divjak, Translation surfaces in the Galilean space, Glas. Mat. Ser. III 46, No. 66, 455-469, 2011.
20. [20] Z.M. Sipus, and B. Divjak, Surfaces of constant curvature in pseudo-Galilean space, Int. J. Math. Sci. 2012, Art. ID 375264, 28 pp, 2012.
21. [21] D. Palman, Drehyzykliden des Galileischen Raumes $G_3$, Math. Pannon. 2 (1), 98-104, 1991.