Skip to main content
SpringerLink
Log in

Cheng–Yau Operator and Gauss Map of Surfaces of Revolution

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

We study the Gauss map G of surfaces of revolution in the 3-dimensional Euclidean space \({{\mathbb {E}}^3}\) with respect to the so-called Cheng–Yau operator \(\square \) acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map G satisfying \(\square G=AG\) for some \(3\times 3\) matrix A are the planes, right circular cones, circular cylinders, and spheres.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alias, Luis J., Gurbuz, N.: An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures. Geom. Dedic. 121, 113–127 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baikoussis, C., Blair, D.E.: On the Gauss map of ruled surfaces. Glasgow Math. J. 34(3), 355–359 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baikoussis, C., Verstraelen, L.: On the Gauss map of helicoidal surfaces. Rend. Sem. Mat. Messina Ser. II 2(16), 31–42 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Chen, B.-Y., Piccinni, P.: Submanifolds with finite type Gauss map. Bull. Austral. Math. Soc. 35(2), 161–186 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225(3), 195–204 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Choi, S.M.: On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space. Tsukuba J. Math. 19(2), 351–367 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Choi, S.M.: On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space. Tsukuba J. Math. 19(2), 285–304 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Choi, S.M., Kim, D.-S., Kim, Y.H., Yoon, D.W.: Circular cone and its Gauss map. Colloq. Math. 129(2), 203–210 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dillen, F., Pas, J., Verstraelen, L.: On the Gauss map of surfaces of revolution. Bull. Inst. Math. Acad. Sin. 18(3), 239–246 (1990)

    MathSciNet  MATH  Google Scholar 

  10. do Carmo, Manfredo P.: Differential geometry of curves and surfaces. Translated from the Portuguese, Prentice-Hall Inc, Englewood Cliffs (1976)

  11. Dursun, U.: Flat surfaces in the Euclidean space \(E^3\) with pointwise 1-type Gauss map. Bull. Malays. Math. Sci. Soc. (2) 33(3), 469–478 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Ki, U.-H., Kim, D.-S., Kim, Y.H., Roh, Y.-M.: Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space. Taiwan. J. Math. 13(1), 317–338 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Kim, D.-S.: On the Gauss map of quadric hypersurfaces. J. Korean Math. Soc. 31(3), 429–437 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Kim, D.-S.: On the Gauss map of hypersurfaces in the space form. J. Korean Math. Soc. 32(3), 509–518 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Kim, D.-S., Kim, Y.H.: Surfaces with planar lines of curvature. Honam Math. J. 32, 777–790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kim, D.-S., Kim, Y.H., Yoon, D.W.: Extended B-scrolls and their Gauss maps. Indian J. Pure Appl. Math. 33(7), 1031–1040 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Kim, D.-S., Song, B.: On the Gauss map of generalized slant cylindrical surfaces. J. Korea Soc. Math. Educ. Ser. B 20(3), 149–158 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Kim, Y.H., Turgay, N.C.: Surfaces in \(E^3\) with \(L_1\)-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 50(3), 935–949 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, Y.H., Yoon, D.W.: On the Gauss map of ruled surfaces in Minkowski space. Rocky Mt. J. Math. 35(5), 1555–1581 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Levi-Civita, T.: Famiglie di superficie isoparametriche nellordinario spacio euclideo. Rend. Acad. Lincei 26, 355–362 (1937)

    MATH  Google Scholar 

  21. Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569–573 (1970)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Dong-Soo Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022926). Young Ho Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042298) and supported by Kyungpook National University Research Fund, 2012.

Author information

Authors and Affiliations

  1. Department of Mathematics, Chonnam National University, Gwangju, 500-757, South Korea

    Dong-Soo Kim

  2. Department of Mathematics, Kunsan National University, Kunsan, 573-701, South Korea

    Jong Ryul Kim

  3. Department of Mathematics, Kyungpook National University, Daegu, 702-701, South Korea

    Young Ho Kim

Authors
  1. Dong-Soo Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jong Ryul Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Young Ho Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Young Ho Kim.

Additional information

Communicated by Young Jin Suh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, DS., 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). https://doi.org/10.1007/s40840-015-0234-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-015-0234-x

Keywords

Mathematics Subject Classification

Search

Navigation