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MERIDIAN SURFACES IN 𝔼4 WITH POINTWISE 1-TYPE GAUSS MAP

  • Arslan, Kadri (Department of Mathematics Uludag University) ;
  • Bulca, Betul (Department of Mathematics Uludag University) ;
  • Milousheva, Velichka (Bulgarian Academy of Sciences Institute of Mathematics and Informatics, "L. Karavelov" Civil Engineering Higher School)
  • Received : 2013.06.10
  • Published : 2014.05.31

Abstract

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.

Keywords

Meridian surfaces;Gauss map;finite type immersions;pointwise 1-type Gauss map

References

  1. K. Arslan, B. K. Bayram, B. Bulca, Y. H. Kim, C. Murathan, and G. Ozturk, Vranceanu Surface in $E^4$ with Pointwise 1-type Gauss map, Indian J. Pure Appl. Math. 42 (2011), no. 1, 41-51. https://doi.org/10.1007/s13226-011-0003-y
  2. 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
  3. C. Baikoussis, B.-Y. Chen, and L. Verstraelen, Ruled surfaces and tubes with finite type Gauss map, Tokyo J. Math. 16 (1993), no. 2, 341-349. https://doi.org/10.3836/tjm/1270128488
  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, Geometry of Submanifolds and its Applications, Science University of Tokyo, 1981.
  6. B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.
  7. B.-Y. Chen, Finite Type Submanifolds and Generalizations, Universita degli Studi di Roma "La Sapienza", Dipartimento di Matematica IV, Rome, 1985.
  8. B.-Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), no. 2, 117-337.
  9. G. Ganchev and V. Milousheva, Invariants and Bonnet-type theorem for surfaces in ${\mathbb{R}}^4$, Cent. Eur. J. Math. 8 (2010), no. 6, 993-1008. https://doi.org/10.2478/s11533-010-0073-9
  10. Y. H. Kim and D. W. Yoon, Ruled surfaces with finite type Gauss map in Minkowski spaces, Soochow J. Math. 26 (2000), no. 1, 85-96.
  11. B.-Y. Chen, M. Choi, and Y. H. Kim, Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. Soc. 42 (2005), no. 3, 447-455. https://doi.org/10.4134/JKMS.2005.42.3.447
  12. B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186. https://doi.org/10.1017/S0004972700013162
  13. 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
  14. M. Choi and Y. H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc. 38 (2001), no. 4, 753-761.
  15. 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
  16. D. A. Yoon, Rotation Surfaces with finite type Gauss map in ${\mathbb{E}}^4$, Indian J. Pura Appl. Math. 32 (2001), no. 12, 1803-1808.
  17. D. A. Yoon, Some properties of the clifford torus as rotation surfaces, Indian J. Pure Appl. Math. 34 (2003), no. 6, 907-915.

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