Geometriae Dedicata

, Volume 30, Issue 1, pp 93–114 | Cite as

Helical geodesic immersions into complex space forms

  • Sadahiro Maeda
  • Yoshihiro Ohnita
Article

Abstract

This paper consists of two parts. One is to construct a class of helical geodesic equivariant immersions of orderd(⩾3), which are neither Kaehler nor totally real immersions, into complex projective spaces. The other is to show the basic results about a helix in complex space forms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bando, S. and Ohnita, Y., ‘Minimal 2-spheres with Constant Curvature inP n(ℂ)’,J. Math. Soc. Japan 39 (1987), 477–488.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Besse, A., ‘Manifolds all of whose Geodesics are Closed’,Ergebnisse der Mathematik, Bd. 93, Springer Berlin, Heiderberg, New York, 1978.Google Scholar
  3. 3.
    Calabi, E., ‘Isometric Imbedding of Complex Manifolds’,Ann. Math. 58 (1953), 1–23.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ferus, D. and Schirrmacher, S., ‘Submanifolds in Euclidean Space with Simple Geodesics’,Math. Ann. 260 (1982), 57–62.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hong, Y., ‘Helical Submanifolds in Euclidean Spaces’,Indiana Univ. Math. J. 35 (1986), 29–43.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Maebashi, T. and Maeda, S., ‘Constant Mean Curvature Submanifolds of Higher Codimensions’,Kodai Math. J. 9 (1986), 175–178.MathSciNetMATHGoogle Scholar
  7. 7.
    Maeda, S., ‘Imbedding of a Complex Projective Space Similar to Segre Imbedding’,Archiv. Math. 37 (1981), 556–560.MATHCrossRefGoogle Scholar
  8. 8.
    Maeda, S. and Sato, N., ‘On Submanifolds all of whose Geodesics are Circles in a Complex Space Form’,Kodai Math. J. 6 (1983), 157–166.MathSciNetMATHGoogle Scholar
  9. 9.
    Naitoh, H., ‘Isotropic Submanifolds with Parallel Second Fundamental Form inP m(c)’,Osaka J. Math. 18 (1981), 427–464.MATHMathSciNetGoogle Scholar
  10. 10.
    Nakagawa, H., and Ogiue, K., ‘Complex Space Forms Immersed in Complex Space Forms’,Trans. Amer. Math. Soc. 219 (1976), 289–297.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ogiue, K., ‘Differential Geometry of Kaehler Submanifolds’,Advances Math. 13 (1974), 73–114.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Ohnita, Y., ‘Homogeneous Harmonic Maps into Complex Projective Spaces’, (Preprint), Max-Planck-Institut für Math., Bonn, 1988.Google Scholar
  13. 13.
    Pak, J. S. and Sakamoto, K., ‘Submanifolds with Properd-Planar Geodesics Immersed in Complex Projective Spaces’,Tohoku Math. J. 38 (1986), 297–311.MathSciNetMATHGoogle Scholar
  14. 14.
    Sakamoto, K., ‘Helical Immersions into a Unit Sphere’,Math. Ann. 261 (1982), 63–80.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Sakamoto, K., ‘Helical Minimal Immersions of Compact Riemannian Manifolds into a Unit Sphere’,Trans. Amer. Math. Soc. 288 (1985), 765–790.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Tsukada, K., ‘Helical Geodesic Immersions of Compact Rank One Symmetric Spaces into Spheres’,Tokyo J. Math. 6 (1983), 267–285.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Sadahiro Maeda
    • 1
  • Yoshihiro Ohnita
    • 2
    • 3
  1. 1.Department of MathematicsKumamoto Institute of TechnologyKumamotoJapan
  2. 2.Max-Planck-Institut für MathematikBonn 3FRG
  3. 3.Department of MathematicsTokyo Metropolitan UniversityTokyoJapan

Personalised recommendations