RULED SURFACES AND GAUSS MAP

Title & Authors
RULED SURFACES AND GAUSS MAP
KIM, DONG-SOO;

Abstract
We study the Gauss map G of ruled surfaces in the 3-dimensional Euclidean space $\small{\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\small{{\Box}}$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only ruled surfaces with Gauss map G satisfying \${\Box}G
Keywords
Gauss map;Cheng-Yau operator;ruled surface;
Language
English
Cited by
References
1.
L. J. Alias and N. Gurbuz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113-127.

2.
C. Baikoussis, Ruled submanifolds with finite type Gauss map, J. Geom. 49 (1994), no. 1-2, 42-45.

3.
C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359.

4.
C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 2(16) (1993), 31-42.

5.
B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publ., New Jersey, 1984.

6.
B.-Y. Chen, Finite Type Submanifolds and Generalizations, University of Rome, 1985.

7.
B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186.

8.
S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204.

9.
M. Choi, D.-S. Kim, Y. H. Kim, and D. W. Yoon, Circular cone and its Gauss map, Colloq. Math. 129 (2012), no. 2, 203-210.

10.
S. M. Choi, On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), no. 2, 351-367.

11.
S. M. Choi, On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), no. 2, 285-304.

12.
F. Dillen, J. Pas, and L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18 (1990), no. 3, 239-246.

13.
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.

14.
U. Dursun, Hypersurfaces with pointwise 1-type Gauss map, Taiwanese J. Math. 11 (2007), no. 5, 1407-1416.

15.
U. Dursun, Flat surfaces in the Euclidean space E3 with pointwise 1-type Gauss map, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 469-478.

16.
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, Taiwanese J. Math. 13 (2009), no. 1, 317-338.

17.
D.-S. Kim, On the Gauss map of quadric hypersurfaces, J. Korean Math. Soc. 31 (1994), no. 3, 429-437.

18.
D.-S. Kim, On the Gauss map of hypersurfaces in the space form, J. Korean Math. Soc. 32 (1995), no. 3, 509-518.

19.
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., To appear. arXiv:1411.2291

20.
D.-S. Kim and Y. H. Kim, Surfaces with planar lines of curvature, Honam Math. J. 32 (2010), no. 4, 777-790.

21.
D.-S. Kim, Y. H. Kim, and D. W. Yoon, Extended B-scrolls and their Gauss maps, Indian J. Pure Appl. Math. 33 (2002), no. 7, 1031-1040.

22.
D.-S. Kim and B. Song, On the Gauss map of generalized slant cylindrical surfaces, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 20 (2013), no. 3, 149-158.

23.
Y. H. Kim and N. C. Turgay, Surfaces in \$E^3\$ with \$L_1\$-pointwise 1-type Gauss map, Bull. Korean Math. Soc. 50 (2013), no. 3, 935-949.

24.
Y. H. Kim and N. C. Turgay, Classifications of helicoidal surfaces with \$L_1\$-pointwise 1-type Gauss map, Bull. Korean Math. Soc. 50 (2013), no. 4, 1345-1356.

25.
Y. H. Kim and D. W. Yoon, On the Gauss map of ruled surfaces in Minkowski space, Rocky Mountain J. Math. 35 (2005), no. 5, 1555-1581.

26.
E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573.