Geometriae Dedicata

, Volume 27, Issue 3, pp 325–334

On totally real surfaces of the nearly Kaehler 6-sphere

  • F. Dillen
  • B. Opozda
  • L. Verstraelen
  • L. Vrancken
Article
  • 56 Downloads

Abstract

Let M be a minimal totally real surface of the nearly Kaehler 6-sphere. We prove that if M is homeomorphic to a sphere, then M is totally geodesic. Consequently, if M is compact and has non-negative Gaussian curvature K, then eithe K=0 or K=1. Finally, we derive from these results that if M has constant Gaussian curvature K, then either K=0 or K=1.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bryant, R. L., ‘Minimal Surfaces of Constant Curvature in SnTrans. Amer. Math. Soc. 290 (1985), 259–271.Google Scholar
  2. 2.
    Calabi, E., ‘Construction and Properties of Some 6-Dimensional Almost Complex Manifolds’, Trans. Amer. Math. Soc. 87 (1958), 407–438.Google Scholar
  3. 3.
    Chern, S. S., ‘On the Minimal Immersions of the Two Sphere in a Space of Constant Curvature’, Problems in Analysis, Princeton Univ. Press, Princeton, N.J., 1970, pp. 27–40.Google Scholar
  4. 4.
    Dillen, F., Opozda, B., Verstraelen, L. and Vrancken, L., ‘On Totally Real 3-Dimensional Submanifolds of the Nearly Kaehler 6-Sphere’, Proc. Amer. Math. Soc. 99 (1987), 741–749.Google Scholar
  5. 5.
    Ejiri, N., ‘Equivariant Minimal Immersions of S2 into S2m(1)’, Trans. Amer. Math. Soc. 297 (1986), 105–124.Google Scholar
  6. 6.
    Fröhlicher, ‘Zur Differentialgeometrie der komplexen Structuren’, Math. Ann. 129 (1955), 50–95.Google Scholar
  7. 7.
    Fukami, F. and Ishihara, S., ‘Almost Hermitian Structure on S6’, Tohoku Math. J. 7 (1955), 151–156.Google Scholar
  8. 8.
    Gray, A., ‘Minimal Varieties and Almost Hermitian Submanifolds’, Michigan Math. J. 12 (1965), 273–287.Google Scholar
  9. 9.
    Lawson, H. B., ‘Local Rigidity Theorems for Minimal Hypersurfaces’, Ann. Math. 89 (1969), 187–197.Google Scholar
  10. 10.
    Mashimo, K., ‘Homogeneous Totally Real Submanifolds of S6’, Tsukuba J. Math. 9 (1985), 185–202.Google Scholar
  11. 11.
    Sekigawa, K., ‘Almost Complex Submanifolds of a 6-Dimensional Sphere’, Kōdai Math. J. 6 (1983), 174–185.Google Scholar

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • F. Dillen
    • 1
  • B. Opozda
    • 2
  • L. Verstraelen
    • 1
  • L. Vrancken
    • 1
  1. 1.Department of MathematicsKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Department of MathematicsUniversity of CracówCracówPoland

Personalised recommendations