Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SPECIAL CLASSES OF MERIDIAN SURFACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE

  • GANCHEV, GEORGI (Institute of Mathematics and Informatics Bulgarian Academy of Sciences) ;
  • MILOUSHEVA, VELICHKA (Institute of Mathematics and Informatics Bulgarian Academy of Sciences)
  • Received : 2014.11.03
  • Published : 2015.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants. In the present paper we give the complete classification of Chen meridian surfaces and meridian surfaces with parallel normal bundle.

Keywords

References

  1. S. Carter and U. Dursun, On generalized Chen and $\kappa$-minimal immersions, Beitrage Algebra Geom. 38 (1997), no. 1, 125-134.
  2. S. Carter and U. Dursun, Partial tubes and Chen submanifolds, J. Geom. 63 (1998), no. 1-2, 30-38. https://doi.org/10.1007/BF01221236
  3. B.-Y. Chen, Geometry of Submanifolds. Marcel Dekker, Inc., New York, 1973.
  4. B.-Y. Chen, Pseudo-Riemannian Geometry, $\delta$-Invariants and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
  5. U. Dursun, On product $\kappa$-Chen submanifolds, Glasgow Math. J. 39 (1997), no. 3, 243- 249. https://doi.org/10.1017/S0017089500032171
  6. U. Dursun, On minimal and Chen immersions in space forms, J. Geom. 66 (1999), no. 1-2, 104-115. https://doi.org/10.1007/BF01225674
  7. G. Ganchev and V. Milousheva, On the theory of surfaces in the four-dimensional Euclidean space, Kodai Math. J. 31 (2008), no. 2, 183-198. https://doi.org/10.2996/kmj/1214442794
  8. 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
  9. O. Garay, Spherical Chen surfaces which are mass-symmetric and of 2-type, J. Geom. 33 (1988), no. 1-2, 39-52. https://doi.org/10.1007/BF01230602
  10. L. Gheysens, P. Verheyen, and L. Verstraelen, Sur les surfaces A ou les surfaces de Chen, C. R. Acad. Sci. Paris, Ser. I Math. 292 (1981), no. 19, 913-916.
  11. L. Gheysens, P. Verheyen, and L. Verstraelen, Characterization and examples of Chen submanifolds, J. Geom. 20 (1983), no. 1, 47-62. https://doi.org/10.1007/BF01917994
  12. C. Houh, On spherical A-submanifolds, Chinese J. Math. 2 (1974), no. 1, 128-134.
  13. E. Iyigun, K. Arslan, and G. Ozturk, A characterization of Chen surfaces in $\mathbb{E}^4$, Bull. Malays. Math. Sci. Soc. 31 (2008), no. 2 209-215.
  14. T. Korpinar and E. Turhan, A new class of time-meridian surfaces of biharmonic S- particles and its Lorentz transformation in Heisenberg spacetime, Int. J. Theor. Phys. 2015 (2015); DOI 10.1007/s10773-015-2621-3.
  15. G. Ozturk, B. Bulca, B. Bayram, and K. Arslan, Meridian surfaces of Weingarten type in 4-dimensional Euclidean space $\mathbb{E}^4$. preprint available at ArXiv:1305.3155.
  16. B. Rouxel, Ruled A-surfaces in Euclidean space $\mathbb{E}^n$, Soochow J. Math. 6 (1980), 117- 121.
  17. B. Rouxel, A-submanifolds in Euclidean space, Kodai Math. J. 4 (1981), no. 1, 181-188. https://doi.org/10.2996/kmj/1138036318
  18. B. Rouxel, Chen submanifolds, Geometry and Topology of Submanifolds, VI, World Sci. Publ., River Edge, NJ, 1994.
  19. L. Verstraelen, A-surfaces with flat normal connection, J. Korean Math. Soc. 15 (1978), no. 1, 1-7.

Cited by

  1. Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric vol.14, pp.2, 2017, https://doi.org/10.1007/s00009-017-0878-x