Surfaces in the Euclidean Space E4 with Pointwise 1-Type Gauss Map

Year 2011, Volume: 40 Issue: 5, 617 - 625, 01.05.2011

Uğur Dursun Güler Gürpınar Arsan

Abstract

Surfaces in the Euclidean Space E4 with Pointwise 1-Type Gauss Map

Abstract

In this article we study surfaces in Euclidean space E 4 with pointwise 1-type Gauss map. We give a characterization of surfaces in E 4 with a pointwise 1-type Gauss map of the first kind. We conclude that an oriented non-minimal surface M in E 4 has a pointwise 1-type Gauss map of the first kind if and only if M is a surface in a 3-sphere of E 4 with constant mean curvature. We also obtain a characterization for non-planar minimal surfaces in E 4 with pointwise 1-type Gauss map of the second kind. Further we give a partial classification of surfaces in E 4 in terms of the pointwise 1-type Gauss map of the second kind.

Keywords

Minimal surface, Normal bundle, Mean curvature, Pointwise 1-type, Gauss map

References

• Arslan, K., Bayram, B. K., Bulca, B., Kim, Y. H., Murathan, C. and ¨Ozt¨urk, G. Rotational embeddings in E4with pointwise 1-type Gauss map, Turk. J. Math. 35, 493–499, 2011.
• Baikoussis, C. and Blair, D. E. On the Gauss map of ruled surfaces, Glasgow Math. J. 34, 355–359, 1992.
• Baikoussis, C., Chen, B. Y. and Verstraelen, L. Ruled surfaces and tubes with finite type Gauss map, Tokyo J. Math. 16, 341–348, 1993.
• Baikoussis, C. Ruled sumanifolds with finite type Gauss map, J. Geom. 49, 42–45, 1994.
• Baikoussis, C. and Verstralen, L. The Chen-type of the spiral surfaces, Results in Math. 28, 214–223, 1995.
• Chen, B. Y. Geometry of Submanifolds (Marcel Dekker, New York, 1973).
• Chen, B. Y. On submanifolds of finite type, Soochow J. Math. 9, 65–81, 1983.
• Chen, B. Y. Total Mean Curvature and Submanifolds of Finite Type (World Scientific, Sin- gapor, New Jersey, London, 1984).
• Chen, B. Y. and Piccinni, P. Sumanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35, 161–186, 1987.
• Chen, B. Y., Choi, M. and Kim, Y. H. Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. 42, 447–455, 2005.
• Choi, M. and Kim, Y. H. Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc. 38, 753–761, 2001.
• Dursun, U. Hypersurfaces with pointwise 1-type Gauss map, Taiwanese J. Math. 11, 1407– 1416, 2007.
• Kim, Y. H. and Yoon, D. W. Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34, 191–205, 2000.
• Kim, Y. H. and Yoon, D. W. Classification of rotation surfaces in pseudo-Euclidean space, J. Korean Math. 41, 379–396, 2004.
• Niang, A. Rotation surfaces with 1-type Gauss map, Bull. Korean Math. Soc. 42, 23–27, 2005.
• Yoon, D. W. Rotation surfaces with finite type Gauss map in E4, Indian J. Pure. Appl. Math. 32, 1803–1808, 2001.
• Yoon, D. W. On the Gauss map of translation surfaces in Minkowski 3-spaces, Taiwanese J. Math. 6, 389–398, 2002.