Browse > Article
http://dx.doi.org/10.4134/BKMS.2015.52.5.1661

RULED SURFACES AND GAUSS MAP  

KIM, DONG-SOO (DEPARTMENT OF MATHEMATICS CHONNAM NATIONAL UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.5, 2015 , pp. 1661-1668 More about this Journal
Abstract
We study the Gauss map G of ruled surfaces in the 3-dimensional Euclidean space $\mathbb{E}^3$ with respect to the so called Cheng-Yau operator ${\Box}$ 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 ${\Box}G=AG$ for some $3{\times}3$ matrix A are the flat ones. Furthermore, we show that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some nonzero $3{\times}3$ matrix A are the cylindrical surfaces.
Keywords
Gauss map; Cheng-Yau operator; ruled surface;
Citations & Related Records
Times Cited By KSCI : 4  (Citation Analysis)
연도 인용수 순위
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.   DOI
3 C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359.   DOI
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.   DOI
8 S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204.   DOI
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.   DOI   ScienceOn
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.   DOI
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.   DOI
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.   DOI
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.   DOI
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 and Y. H. Kim, Surfaces with planar lines of curvature, Honam Math. J. 32 (2010), no. 4, 777-790.   DOI   ScienceOn
20 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
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI
26 E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573.   DOI   ScienceOn