DOI QR코드

DOI QR Code

NON-ZERO CONSTANT CURVATURE FACTORABLE SURFACES IN PSEUDO-GALILEAN SPACE

  • Received : 2017.03.03
  • Accepted : 2017.06.29
  • Published : 2018.01.31

Abstract

Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces in differential geometry. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [2]. In this study, we provide new results relating to the factorable surfaces with non-zero constant Gaussian and mean curvature.

Keywords

References

  1. M. E. Aydin, A. Mihai, A. O. Ogrenmis, and M. Ergut, Geometry of the solutions of localized induction equation in the pseudo-Galilean space, Adv. Math. Phys. 2015 (2015), Art ID905978, 7pp.
  2. M. E. Aydin, A. O. Ogrenmis, and M. Ergut, Classification of factorable surfaces in the pseudo-Galilean space, Glas. Mat. Ser. III 50(70) (2015), no. 2, 441-451. https://doi.org/10.3336/gm.50.2.12
  3. M. Bekkar and B. Senoussi, Factorable surfaces in the three-dimensional Euclidean and Lorentzian spaces satisfying ${\Delta}r_i\;=\;{\lambda}_ir_i$, J. Geom. 103 (2012), no. 1, 17-29. https://doi.org/10.1007/s00022-012-0117-3
  4. B.-Y. Chen, Pseudo-Riemannian Geometry, $\delta$-Invariants and Applications, World Sci-entific, Hackensack, NJ, 2011.
  5. B.-Y. Chen, A note on homogeneous production models, Kragujevac J. Math. 36 (2012), no. 1, 41-43.
  6. B.-Y. Chen, Solutions to homogeneous Monge-Ampere equations of homothetic functions and their applications to production models in economics, J. Math. Anal. Appl. 411 (2014), no. 1, 223-229. https://doi.org/10.1016/j.jmaa.2013.09.029
  7. B.-Y. Chen and G. E. Vilcu, Geometric classifications of homogeneous production functions, Appl. Math. Comput. 225 (2013), 345-351.
  8. M. J. P. Cullen and R. J. Douglas, Applications of the Monge-Ampere equation and Monge transport problem to meterology and oceanography, In: L. A. Caffarelli, M. Mil-man (eds.), NSF-CBMS Conference on the Monge Ampere Equation, Applications to Geometry and Optimization, July 9-13, pp. 33-54, Florida Atlantic University, 1997.
  9. M. Dede, Tube surfaces in pseudo-Galilean space, Int. J. Geom. Methods Mod. Phys. 13 (2016), no. 5, 1650056, 10 pp.
  10. B. Divjak and Z. M. Sipus, Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces, Acta Math. Hungar. 98 (2003), no. 3, 203-215. https://doi.org/10.1023/A:1022821824927
  11. Z. Erjavec, On generalization of helices in the Galilean and the pseudo-Galilean space, J. Math. Research 6 (2014), no. 3, 39-50.
  12. O. Giering, Vorlesungen uber hohere Geometrie, Friedr. Vieweg & Sohn, Braunschweig, Germany, 1982.
  13. W. Goemans and I. Van de Woestyne, Translation and homothetical lightlike hypersur-faces of semi-Euclidean space, Kuwait J. Sci. Engrg. 38 (2011), no. 2A, 35-42.
  14. A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press LLC, 1998.
  15. L. Jiu and H. Sun, On minimal homothetical hypersurfaces, Colloq. Math. 109 (2007), no. 2, 239-249. https://doi.org/10.4064/cm109-2-6
  16. D. Klawitter, Clifford Algebras: Geometric Modelling and Chain Geometries with Ap-plication in Kinematics, Springer Spektrum, 2015.
  17. R. Lopez, Separation of variables in equations of mean-curvature type, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 5, 1017-1035. https://doi.org/10.1017/S0308210515000815
  18. R. Lopez and M. Moruz, Translation and homothetical surfaces in Euclidean space with constant curvature, J. Korean Math. Soc. 52 (2015), no. 3, 523-535. https://doi.org/10.4134/JKMS.2015.52.3.523
  19. H. Meng and H. Liu, Factorable surfaces in Minkowski space, Bull. Korean Math. Soc. 46 (2009), no. 1, 155-169. https://doi.org/10.4134/BKMS.2009.46.1.155
  20. Z. Milin-Sipus, On a certain class of translation surfaces in a pseudo-Galilean space, Int. Mat. Forum 6 (2012), no. 23, 1113-1125.
  21. Z. Milin-Sipus and B. Divjak, Some special surfaces in the pseudo-Galilean Space, Acta Math. Hungar. 118 (2008), no. 3, 209-226. https://doi.org/10.1007/s10474-007-6171-x
  22. Z. Milin-Sipus and B. Divjak, Surfaces of constant curvature in the pseudo-Galilean space, Int. J. Math. Sci. 2012 (2012), Art ID375264, 28pp.
  23. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  24. A. Onishchick and R. Sulanke, Projective and Cayley-Klein Geometries, Springer, 2006.
  25. A. D. Polyanin, W. E. Schiesser, and A. I. Zhurov, Partial differential equation, Schol-arpedia, 3 (2008), no. 10, 4605, revision #121514.
  26. L. Simon, The minimal surface equation, Geometry, V, 239-272, Encyclopaedia Math. Sci., 90, Springer, Berlin, 1997.
  27. L. Simon, Equations of mean curvature type in 2 independent variables, Pacific J. Math. 69 (1977), no. 1, 245-268. https://doi.org/10.2140/pjm.1977.69.245
  28. V. Ushakov, The explicit general solution of trivial Monge-Ampere equation, Comment. Math. Helv. 75 (2000), no. 1, 125-133. https://doi.org/10.1007/s000140050115
  29. I. Van de Woestyne, Minimal homothetical hypersurfaces of a semi-Euclidean space, Results Math. 27 (1995), no. 3-4, 333-342. https://doi.org/10.1007/BF03322837
  30. I. M. Yaglom, A simple non-Euclidean Geometry and Its Physical Basis, An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library. Translated from the Russian by Abe Shenitzer. With the editorial assistance of Basil Gordon. Springer-Verlag, New York-Heidelberg, 1979.
  31. D. W. Yoon, Classification of rotational surfaces in pseudo-Galilean space, Glas. Mat. Ser. III 50 (2015), no. 2, 453-465. https://doi.org/10.3336/gm.50.2.13
  32. Y. Yu and H. Liu, The factorable minimal surfaces, Proceedings of the Eleventh In-ternational Workshop on Differential Geometry, 33-39, Kyungpook Nat. Univ., Taegu, 2007.
  33. P. Zong, L. Xiao, and H. Liu, Affine factorable surfaces in three-dimensional Euclidean space, Acta Math. Sinica (Chin. Ser.) 58 (2015), no. 2, 329-336.