Annals of Global Analysis and Geometry

, Volume 30, Issue 4, pp 383–396 | Cite as

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

Original Research
First Online:
Received:
Accepted:

Abstract

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.

Key words

CR submanifold complex Euclidean space second fundamental form almost contact metric structure 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bejancu, A.: CR-submanifolds of a Kähler manifold I, Proc. Amer. Math. Soc., 69, 135–142 (1978)MATHCrossRefMathSciNetGoogle Scholar
  2. Blair, D.E.: Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509 Springer, Berlin (1976)Google Scholar
  3. Cartan, E.: Sur quelques familles remarquables d'hypersurfaces, C. R. Congrés Math. Liége 30–41 (1939); Oeuvres completes Tom III. vol. 2, 1481–1492Google Scholar
  4. Cecil, T.E. and Ryan, P.J.: Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269, 481–499 (1982)MATHCrossRefMathSciNetGoogle Scholar
  5. Chen, B.Y.: Geometry of submanifolds, Pure Appl. Math. 22, Marcel Dekker, New York (1973)Google Scholar
  6. Djorić, M. and Okumura, M.: On contact submanifolds in complex projective space, Math. Nachr., 202, 17–28 (1999)MathSciNetGoogle Scholar
  7. Djorić, M. and Okumura, M.: CR submanifolds of maximal CR dimension of complex projective space, Arch. Math., 71, 148–158 (1998)CrossRefGoogle Scholar
  8. Djorić, M. and Okumura, M.: CR submanifolds of maximal CR dimension in complex manifolds, PDE's, Submanifolds and Affine Differential Geometry, Banach center publications, Institute of Mathematics, Polish Academy of Sciences, Warsaw 2002., 57, 89–99 (2002)Google Scholar
  9. Djorić, M. and Okumura, M.: CR submanifolds of maximal CR dimension in complex space forms and second fundamental form, Proceedings of the Workshop Contemporary Geometry and Related Topics, Belgrade, May 15–21, 2002, 105–116 (2004)Google Scholar
  10. Djorić, M. and Okumura, M.: Certain CR submanifolds of maximal CR dimension of complex space forms, submittedGoogle Scholar
  11. Djorić, M. and Okumura, M.: Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex projective space, submittedGoogle Scholar
  12. Erbacher, J.: Reduction of the codimension of an isometric immersion, J. Differential Geom., 5, 333–340 (1971)MATHMathSciNetGoogle Scholar
  13. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry II, Interscience, New York (1969)Google Scholar
  14. Kon, M.: Pseudo-Einstein real hypersurfaces in complex space forms, J. Differential Geom. 14, 339–354 (1979)MATHMathSciNetGoogle Scholar
  15. Montiel, S. and Romero, A.: On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata, 20, 245–261 (1986)MATHCrossRefMathSciNetGoogle Scholar
  16. Niebergall, R. and Ryan, P.J.: Real hypersurfaces in complex space forms, in Tight and taut submanifolds, (eds. T.E. Cecil and S.-s. Chern), Math. Sci. Res. Inst. Publ. 32, Cambridge Univ. Press, Cambridge, 233–305 (1997)Google Scholar
  17. Nirenberg, R. and Wells, R.O. Jr.: Approximation theorems on differentiable submanifolds of a complex manifold, Trans. Amer. Math. Soc., 142, 15–35 (1965)CrossRefMathSciNetGoogle Scholar
  18. Olszak, Z.: Contact metric hypersurfaces in complex spaces, Demonstratio Math., 16, 95–102 (1983)MATHMathSciNetGoogle Scholar
  19. Okumura, M.: On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc., 212, 355–364 (1975)MATHCrossRefMathSciNetGoogle Scholar
  20. Okumura, M.: Contact hypersurfaces in certain Kaehlerian manifolds, Tôhoku Math. J., 18, 74–102 (1966)MATHCrossRefMathSciNetGoogle Scholar
  21. Okumura, M. and Vanhecke, L.: A class of normal almost contact CR-submanifolds in Cq, Rend. Sem. Mat. Univ. Polotec. Torino, 52, 359–369 (1994)MATHMathSciNetGoogle Scholar
  22. Ryan, P.J.: Homogeneity and some curvature conditions for hypersurfaces, Tôhoku Math. J. 21, 363–388 (1969)Google Scholar
  23. Tashiro, Y.: On contact structure of hypersurfaces in complex manifold I, Tôhoku Math. J., 15, 62–78 (1963)MATHCrossRefMathSciNetGoogle Scholar
  24. Tashiro, Y.: Relations between almost complex spaces and almost contact spaces, Sûgaku, 16, 34–61 (1964)MathSciNetGoogle Scholar
  25. Tashiro, Y. and Tachibana, S.: On Fubinian and C-Fubinian manifolds, Kōdai Math.Sem.Rep., 15, 176–183 (1963)MATHCrossRefMathSciNetGoogle Scholar
  26. Tumanov, A.E.: The geometry of CR manifolds, Encyclopedia of Math. Sci. 9 VI, Several complex variables III, Springer-Verlag, 201–221 (1986)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeMontenegro
  2. 2.5-25-25, Minami Ikuta, Tama-kuKawasakiJapan

Personalised recommendations