DOI QR코드

DOI QR Code

TENSOR PRODUCT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

  • Arslan, Kadri ;
  • Bulca, Betul ;
  • Kilic, Bengu ;
  • Kim, Young-Ho ;
  • Murathan, Cengizhan ;
  • Ozturk, Gunay
  • Received : 2009.10.07
  • Published : 2011.05.31

Abstract

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ to have pointwise 1-type Gauss map.

Keywords

tensor product immersion;Gauss map;finite type;pointwise 1-type

References

  1. 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
  2. C. Baikoussis, B. Y. Chen, and L. Verstraelen, Ruled surfaces and tubes with nite type Gauss map, Tokyo J. Math. 16 (1993), no. 2, 341-349. https://doi.org/10.3836/tjm/1270128488
  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. B.-Y. Chen, Geometry of Submanifolds and Its Applications, Science University of Tokyo, Tokyo, 1981.
  5. B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.
  6. B.-Y. Chen, Finite Type Submanifolds and Generalizations, Universita degli Studi di Roma "La Sapienza", Istituto Matematico "Guido Castelnuovo", Rome, 1985.
  7. B.-Y. Chen, Differential geometry of semiring of immersions. I. General theory, Bull. Inst. Math. Acad. Sinica 21 (1993), no. 1, 1-34.
  8. 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
  9. B.-Y. Chen and P. Piccinni, Submanifolds with nite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186. https://doi.org/10.1017/S0004972700013162
  10. 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.
  11. F. Decruyenaere, F. Dillen, I. Mihai, and L. Verstraelen, Tensor products of spherical and equivariant immersions, Bull. Belg. Math. Soc. Simon Stevin 1 (1994), no. 5, 643- 648.
  12. F. Decruyenaere, F. Dillen, L. Verstraelen, and L. Vrancken, The semiring of immersions of manifolds, Beitrage Algebra Geom. 34 (1993), no. 2, 209-215.
  13. I. Mihai, R. Rosca, L. Verstraelen, and L. Vrancken, Tensor product surfaces of Euclidean planar curves, Rend. Sem. Mat. Messina Ser. II 3(18) (1994/95), 173-185.
  14. 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.
  15. 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
  16. 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
  17. D. A. Yoon, Rotation surfaces with finite type Gauss map in E4, Indian J. Pure Appl. Math. 32 (2001), no. 12, 1803-1808.
  18. D. A. Yoon, Some properties of the Clifford torus as rotation surfaces, Indian J. Pure Appl. Math. 34 (2003), no. 6, 907-915.

Cited by

  1. General rotational surfaces with pointwise 1-type Gauss map in pseudo-Euclidean space E 2 4 vol.46, pp.1, 2015, https://doi.org/10.1007/s13226-015-0112-0
  2. A study on tensor product surfaces in low-dimensional Euclidean spaces vol.64, pp.12, 2013, https://doi.org/10.1007/s11253-013-0755-0
  3. BOOST INVARIANT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN MINKOWSKI 4-SPACE E41 vol.51, pp.6, 2014, https://doi.org/10.4134/BKMS.2014.51.6.1863
  4. FLAT ROTATIONAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN E4 vol.38, pp.2, 2016, https://doi.org/10.5831/HMJ.2016.38.2.305