# ON THE GAUSS MAP OF SURFACES OF REVOLUTION WITHOUT PARABOLIC POINTS

• Kim, Young-Ho (Department of Mathematics Teachers' College Kyungpook National University) ;
• Lee, Chul-Woo (Department of Mathematics College of Natural Sciences Kyungpook National University) ;
• Yoon, Dae-Won (Department of Mathematics Education and RINS Gyeongsang National University)
• Published : 2009.11.30

#### Abstract

In this article, we study surfaces of revolution without parabolic points in a Euclidean 3-space whose Gauss map G satisfies the condition ${\Delta}^hG\;=\;AG,A\;{\in}\;Mat(3,{\mathbb{R}}),\;where\;{\Delta}^h$ denotes the Laplace operator of the second fundamental form h of the surface and Mat(3,$\mathbb{R}$) the set of 3${\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.

#### 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 https://doi.org/10.21099/tkbjm/1496161968
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 https://doi.org/10.21099/tkbjm/1496163588
3. C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 2(16) (1993), 31-42
4. C. Baikoussis and L. Verstraelen, The Chen-type of the spiral surfaces, Results Math. 28 (1995), no. 3-4, 214-223 https://doi.org/10.1007/BF03322254
5. B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), no. 2, 117-337
6. 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 https://doi.org/10.21099/tkbjm/1496162874
7. 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
8. 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
9. 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 https://doi.org/10.4134/JKMS.2004.41.2.379
10. 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 https://doi.org/10.1016/S0393-0440(99)00063-7
11. 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 https://doi.org/10.1216/rmjm/1181069651
12. E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573 https://doi.org/10.2307/1995413
13. D. W. Yoon, Rotation surfaces with finite type Gauss map in $E^4$, Indian J. Pure Appl. Math. 32 (2001), no. 12, 1803-1808
14. C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359 https://doi.org/10.1017/S0017089500008946
15. 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
16. D. W. Yoon, On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese J. Math. 6 (2002), no. 3, 389-398 https://doi.org/10.11650/twjm/1500558304

#### Cited by

1. SURFACES OF REVOLUTION SATISFYING ΔIIG = f(G + C) vol.50, pp.4, 2013, https://doi.org/10.4134/BKMS.2013.50.4.1061
2. On the Gauss Map of Surfaces of Revolution with Lightlike Axis in Minkowski 3-Space vol.2013, 2013, https://doi.org/10.1155/2013/130495
3. Classification of rotational surfaces in pseudo-Galilean space vol.50, pp.2, 2015, https://doi.org/10.3336/gm.50.2.13
4. SURFACES OF REVOLUTION WITH LIGHT-LIKE AXIS vol.25, pp.4, 2012, https://doi.org/10.14403/jcms.2012.25.4.677