Advertisement

Cheng–Yau Operator and Gauss Map of Rotational Hypersurfaces in 4-Space

  • Erhan GülerEmail author
  • Nurettin Cenk Turgay
Article
  • 26 Downloads

Abstract

We consider rotational hypersurface in the four-dimensional Euclidean space \( {\mathbb {E}}^{4}\). We study the Gauss map \(\mathbf {G}\) of rotational hypersurface in \({\mathbb {E}}^{4}\) with respect to the so-called Cheng–Yau operator \(L_{1}\) acting on the functions defined on the hypersurfaces. We obtain the classification theorem that the only rotational hypersurface with Gauss map \(\mathbf {G}\) satisfying \(L_{1}\mathbf {G}=\mathbf {AG}\) for some \( 4\times 4\) matrix \(\mathbf {A}\) are the hyperplanes, right circular hypercones, circular hypercylinders, and hyperspheres.

Keywords

Euclidean spaces Cheng–Yau operator finite type mappings rotational hypersurfaces \(L_{k}\)-operators 

Mathematics Subject Classification

Primary 53A35 Secondary 53C42 

Notes

References

  1. 1.
    Alias, L.J., Gürbüz, N.: An extension of Takashi theorem for the linearized operators of the highest order mean curvatures. Geom. Dedicata 121, 113–127 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arslan, K., Deszcz, R., Yaprak, S.: On Weyl pseudosymmetric hypersurfaces. Colloq. Math. 72(2), 353–361 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arvanitoyeorgos, A., Kaimakamis, G., Magid, M.: Lorentz hypersurfaces in \({\mathbb{E}}_{1}^{4}\) satisfying \(\Delta H=\alpha H\). Ill. J. Math. 53(2), 581–590 (2009)CrossRefGoogle Scholar
  4. 4.
    Bour, E.: Théorie de la déformation des surfaces. J. de l.Ecole Imperiale Polytechnique 22(39), 1–148 (1862)Google Scholar
  5. 5.
    Chen, B.Y.: Total mean curvature and submanifolds of finite type. World Scientific, Singapore (1984)CrossRefGoogle Scholar
  6. 6.
    Cheng, Q.M., Wan, Q.R.: Complete hypersurfaces of \({\mathbb{R}} ^{4}\) with constant mean curvature. Monatsh. Math. 118, 171–204 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Choi, M., Kim, Y.H.: Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 38, 753–761 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dillen, F., Pas, J., Verstraelen, L.: On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 13, 10–21 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Do Carmo, M., Dajczer, M.: Helicoidal surfaces with constant mean curvature. Tohoku Math. J. 34, 351–367 (1982)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Do Carmo, M., Dajczer, M.: Rotation hypersurfaces in spaces of constant curvature. Trans. Am. Math. Soc. 277, 685–709 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ferràndez, A., Garay, O.J., Lucas, P.: On a certain class of conformally flat Euclidean hypersurfaces. Global differential geometry and global analysis (Berlin, 1990), pp. 48–54. Lecture Notes in Math., 1481, Springer, Berlin (1991)CrossRefGoogle Scholar
  13. 13.
    Ganchev, G., Milousheva, V.: General rotational surfaces in the 4-dimensional Minkowski space. Turk. J. Math. 38, 883–895 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Güler, E., Hacisalihoglu, H.H., Kim, Y.H.: The Gauss map and the third Laplace–Beltrami operator of the rotational hypersurface in 4-space. Symmetry 10(9), 1–12 (2018)CrossRefGoogle Scholar
  15. 15.
    Güler, E., Kaimakamis, G., Magid, M.: Helicoidal hypersurfaces in Minkowski 4-space \({\mathbb{E}}_{1}^{4}\) (under review) Google Scholar
  16. 16.
    Güler, E., Magid, M., Yaylı, Y.: Laplace-Beltrami operator of a helicoidal hypersurface in four-space. J. Geom. Symmetry Phys. 41, 77–95 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kim, D.S., Kim, J.R., Kim, Y.H.: Cheng–Yau operator and Gauss map of surfaces of revolution. Bull. Malays. Math. Sci. Soc. 39, 1319–1327 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kim, Y.H., Turgay, N.C.: Surfaces in \({\mathbb{E}}^{4}\) with \(L_{1}\)-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 50(3), 935–949 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lawson, H.B.: Lectures on minimal submanifolds. 2nd ed.; Mathematics Lecture Series 9. Publish or Perish, Inc., Wilmington (1980)Google Scholar
  20. 20.
    Levi-Civita, T.: Famiglie di superficie isoparametriche nellordinario spacio euclideo. Rend. Acad. Lincei 26, 355–362 (1937)zbMATHGoogle Scholar
  21. 21.
    Magid, M., Scharlach, C., Vrancken, L.: Affine umbilical surfaces in \({\mathbb{R}}^{4}\). Manuscripta Math. 88, 275–289 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Moore, C.: Surfaces of rotation in a space of four dimensions. Ann. Math. 21, 81–93 (1919)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Moore, C.: Rotation surfaces of constant curvature in space of four dimensions. Bull. Am. Math. Soc. 26, 454–460 (1920)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Moruz, M., Munteanu, M.I.: Minimal translation hypersurfaces in \({\mathbb{E}}^{4}\). J. Math. Anal. Appl. 439, 798–812 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Scharlach, C.: Affine geometry of surfaces and hypersurfaces in \( {\mathbb{R}}^{4}\). In: Dillen, F., Simon, U., Vrancken, L. (eds.) Symposium on the Differential Geometry of Submanifolds, pp. 251–256. Un. Valenciennes, Valenciennes (2007)Google Scholar
  26. 26.
    Senoussi, B., Bekkar, M.: Helicoidal surfaces with \(\Delta ^{J}r=Ar\) in 3-dimensional Euclidean space. Stud. Univ. Babeş-Bolyai Math. 60(3), 437–448 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 18, 380–385 (1966)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Verstraelen, L., Valrave, J., Yaprak, S.: The minimal translation surfaces in Euclidean space. Soochow J. Math. 20(1), 77–82 (1994)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Vlachos, Th: Hypersurfaces in \({\mathbb{E}}^{4}\) with harmonic mean curvature vector field. Math. Nachr. 172, 145–169 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics Faculty of SciencesBartın UniversityBartınTurkey
  2. 2.Department of Mathematics Faculty of Science and LettersIstanbul Technical UniversityIstanbulTurkey

Personalised recommendations