DOI QR코드

DOI QR Code

HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE

  • Choi, Mie-Kyung (DEPARTMENT OF MATHEMATICS KYUNGPOOK NATIONAL UNIVERSITY) ;
  • Kim, Young-Ho (DEPARTMENT OF MATHEMATICS KYUNGPOOK NATIONAL UNIVERSITY) ;
  • Liu, Huili (DEPARTMENT OF MATHEMATICS NORTHEASTERN UNIVERSITY) ;
  • Yoon, Dae-Won (DEPARTMENT OF MATHEMATICS EDUCATION AND RINS GYEONGSANG NATIONAL UNIVERSITY)
  • Received : 2009.03.10
  • Published : 2010.07.31

Abstract

The helicoidal surface is a generalization of rotation surface in a Minkowski space. We study helicoidal surfaces in a Minkowski 3-space in terms of their Gauss map and provide some examples of new classes of helicoidal surfaces with constant mean curvature in a Minkowski 3-space.

Keywords

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 and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 2(16) (1993), 31–42.
  3. C. C. Beneki and T. Koufogiorgos, Helicoidal surfaces with prescribed mean or Gaussian curvature, J. Geom. 63 (1998), no. 1-2, 25–29. https://doi.org/10.1007/BF01221235
  4. C. C. Beneki, G. Kaimakamis, and B. J. Papantoniou, Helicoidal surfaces in threedimensional Minkowski space, J. Math. Anal. Appl. 275 (2002), no. 2, 586–614. https://doi.org/10.1016/S0022-247X(02)00269-X
  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, University of Rome, 1985.
  7. 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
  8. 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
  9. 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.
  10. F. Ji and Z. H. Hou, Helicoidal surfaces under the cubic screw motion in Minkowski 3-space, J. Math. Anal. Appl. 318 (2006), no. 2, 634–647. https://doi.org/10.1016/j.jmaa.2005.06.032
  11. U.-H. Ki, D.-S. Kim, Y. H. Kim, and Y.-M. Roh, Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space, Taiwanese J. Math. 13 (2009), no. 1, 317–338.
  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. https://doi.org/10.1016/S0393-0440(99)00063-7
  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. https://doi.org/10.1216/rmjm/1181069651

Cited by

  1. SHAPE OPERATOR AND GAUSS MAP OF POINTWISE 1-TYPE vol.52, pp.6, 2015, https://doi.org/10.4134/JKMS.2015.52.6.1337
  2. CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP vol.50, pp.4, 2013, https://doi.org/10.4134/BKMS.2013.50.4.1345
  3. Classifications of Canal Surfaces with L1-Pointwise 1-Type Gauss Map vol.83, pp.1, 2015, https://doi.org/10.1007/s00032-015-0233-2
  4. HELICOIDAL SURFACES OF THE THIRD FUNDAMENTAL FORM IN MINKOWSKI 3-SPACE vol.52, pp.5, 2015, https://doi.org/10.4134/BKMS.2015.52.5.1569
  5. The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space vol.10, pp.9, 2018, https://doi.org/10.3390/sym10090398