ON THE GAUSS MAP OF SURFACES OF REVOLUTION WITHOUT PARABOLIC POINTS

Title & Authors
ON THE GAUSS MAP OF SURFACES OF REVOLUTION WITHOUT PARABOLIC POINTS
Kim, Young-Ho; Lee, Chul-Woo; Yoon, Dae-Won;

Abstract
In this article, we study surfaces of revolution without parabolic points in a Euclidean 3-space whose Gauss map G satisfies the condition $\small{{\Delta}^hG\;=\;AG,A\;{\in}\;Mat(3,{\mathbb{R}}),\;where\;{\Delta}^h}$ denotes the Laplace operator of the second fundamental form h of the surface and Mat(3,$\small{\mathbb{R}}$) the set of 3$\small{{\times}}$3-real matrices, and also obtain the complete classification theorem for those. In particular, we have a characterization of an ordinary sphere in terms of it.
Keywords
Gauss map;surface of revolution;Laplace operator;second fundamental form;
Language
English
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대한수학회보, 2013. vol.50. 4, pp.1061-1067
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References
1.
R. Aiyama, On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space, Tsukuba J. Math. 16 (1992), no. 2, 353-361

2.
L. J. Alias, A. Ferr´andez, P. Lucas, and M. A. Mero.no, On the Gauss map of B-scrolls, Tsukuba J. Math. 22 (1998), no. 2, 371-377

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.
C. Baikoussis and L. Verstraelen, The Chen-type of the spiral surfaces, Results Math. 28 (1995), no. 3-4, 214-223

6.
B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), no. 2, 117-337

7.
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

8.
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

9.
F. Dillen, J. Pas, and L. Verstralen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18 (1990), no. 3, 239-246

10.
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

11.
Y. H. Kim and D. W. Yoon, Classifications of rotation surfaces in pseudo-Euclidean space, J. Korean Math. Soc. 41 (2004), no. 2, 379-396

12.
Y. H. Kim and D. W. Yoon, Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (2000), no. 3-4, 191-205

13.
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

14.
E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573

15.
D. W. Yoon, On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese J. Math. 6 (2002), no. 3, 389-398

16.
D. W. Yoon, Rotation surfaces with finite type Gauss map in $E^4$, Indian J. Pure Appl. Math. 32 (2001), no. 12, 1803-1808