Advertisement

Geometriae Dedicata

, Volume 62, Issue 2, pp 115–138 | Cite as

Submanifolds with totally umbilical Gauss image

  • Yury A. Nikolayevsky
Article
  • 31 Downloads

Abstract

The submanifolds whose Gauss images are totally umbilical submanifolds of the Grassmann manifold are under consideration. The main result is the following classification theorem: if the Gauss image of a submanifold F in a Euclidean space is totally umbilical then either the Gauss image is totally geodesic, or F is the surface in E4 of the special structure. Submanifolds in a Euclidean space with totally geodesic Gauss image were classified earlier.

Mathematics Subject Classifications (1991)

53A07 53B25 

Key words

Gauss image totally umbilical submanifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nikolayevsky, Yu. A.: Submanifolds of Euclidean space with totally geodesic Gauss image, Soviet Math. Dokl. 314 (1990), 296–300.Google Scholar
  2. 2.
    Nikolayevsky, Yu. A.: Classification of the multidimensional submanifolds in a Euclidean space with totally geodesic Gauss image, Math. USSR Sb. 183 (1992), 127–154.Google Scholar
  3. 3.
    Ferus, D.: Symmetric submanifolds of Euclidean space, Math. Ann. 247 (1980), 81–93.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kim, J. J.: Submanifolds with totally umbilical Gauss image immersed in a Euclidean space, Kyungpook Math. J. 27 (1987), 15–26.zbMATHMathSciNetGoogle Scholar
  5. 5.
    Itoh, T.: Isotropic submanifolds in a Euclidean space, Proc. Japan Acad. A62 (1986), 382–385.MathSciNetGoogle Scholar
  6. 6.
    Ki, U.-H. and Pak, J. S.: Submanifolds of a Euclidean m-space E m with totally umbilical Gauss image, Tensor 44 (1987), 233–239.MathSciNetGoogle Scholar
  7. 7.
    Pak, J. S. and Kim, J. J.: Isotropic immersions with totally geodesic Gauss image, Tensor 43 (1987), 167–174.MathSciNetGoogle Scholar
  8. 8.
    Sakamoto, K.: Planar geodesic immersions, Tohoku Math. J. 29 (1977), 25–56.zbMATHMathSciNetGoogle Scholar
  9. 9.
    Nikolayevsky, Yu. A.: Totally umbilical submanifolds of G(2, n). I, Ukr. Geom. Sb. 34 (1991), 83–98.Google Scholar
  10. 10.
    Nikolayevsky, Yu. A.: Totally umbilical submanifolds in symmetric spaces, Math. Phys. Anal. Geom. 1 (1994), 314–357.Google Scholar
  11. 11.
    Helgason, S.: Differential geometry and symmetric spaces, Academic Press, 1962.Google Scholar
  12. 12.
    Hoffman, D. and Osserman, R.: The geometry of the generalized Gauss map, Mem. Amer. Math. Soc. 28 (1980), 1–150.MathSciNetGoogle Scholar
  13. 13.
    Aminov, Yu, A.: Determination of the surface in the 4-dimensional Euclidean space by its Gauss image, Math. USSR Sb. 117 (1982), 147–160.zbMATHMathSciNetGoogle Scholar
  14. 14.
    Borisenko, A.A.: On the unique determination of the multi-dimensional submanifolds in a Euclidean space by its Gauss image, Math. Notes 51 (1992), 8–15.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Chen, B.Y. and Nagano, T.: Totally geodesic submanifolds of symmetric spaces. I, Duke Math. J. 44 (1977), 745–755.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Lankaster, P.: Theory of Matrices, Academic Press, New York 1969.Google Scholar
  17. 17.
    Nikolayevsky, Yu. A.: Totally umbilical submanifolds of G(2, n). II, III, IV, Math. Phys., Anal., Geom. (to appear).Google Scholar
  18. 18.
    Erbacher, J. A.: Reduction of the codimension of an isometric immersion, J. Differential Geom. 5 (1971), 333–340.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Chen, B.-Y.: Geometry of Submanifolds, Marcel Dekker, New York 1973.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Yury A. Nikolayevsky
    • 1
  1. 1.Department of MathematicsMelbourne UniversityVictoriaAustralia

Personalised recommendations