Geometriae Dedicata

, Volume 90, Issue 1, pp 115–125 | Cite as

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

  • Xu Cheng


In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.

Kähler manifold coisotropic submanifold holomorphic curvature 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BC] Bishop, R. L. and Crittenden, R. J.: Geometry of Manifolds, Academic Press, New York, 1964.Google Scholar
  2. [F1] Frankel, T.: Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174.Google Scholar
  3. [F2] Frankel, T.: On the fundamental group of a compact minimal submanifold, Ann. Math. 83 (1966), 68–73.Google Scholar
  4. [Ga] Galloway, G. J.: Some results on the occurrences of compact minimal submanifolds, Manuscripta Math. 25 (1981), 209–219.Google Scholar
  5. [GaR] Galloway, G. J. and Rodriguez, L.: Intersection of minimal submanifolds, Geom. Dedicata 39 (1991), 29–42.Google Scholar
  6. [Go] Goldberg, S. I.: Curvature and Homology, Academic Press, New York, 1962.Google Scholar
  7. [GoK] Goldberg, S. I. and Kobayashi, S.: Holomorphic bisectional curvature, J. Differential Geom. 1 (1966), 225–233.Google Scholar
  8. [Gr] Gray, A.: Nearly Kähler manifolds, J. Differential Geom. 4 (1970), 283–309.Google Scholar
  9. [KX] Kenmotsu, K. and Xia, C. Y.: Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific J. Math. 176 (1996), 129–139.Google Scholar
  10. [S] Sakai, T.: Three remarks on fundamental groups of some Riemannian manifolds, Tohoku Math. J. 22 (1970), 249–253.Google Scholar
  11. [Sy1] Synge, J. L.: The first and second variations of length in Riemannian space, Proc. London Math. Soc. 25(1926).Google Scholar
  12. [Sy2] Synge, J. L.: On the connectivity of spaces of positive curvature, Quart. J. Math. (Oxford series) 7 (1936), 316–320.Google Scholar
  13. [T] Tipler, F. J.: General relativity and conjugate ordinary differential equations, J. DifferentialEq uations 30 (1978), 165–174.Google Scholar
  14. [Ts] Tsukamoto, Y.: On Kählerian manifolds with positive holomorphic curvature, Proc. Japan Acad. 33 (1957), 333–335.Google Scholar
  15. [W1] Weinstein, A.: Lectures on SymplecticManifolds, CBMS Regional Conf. Ser. Math., 29, Amer. Math. Soc., Providence, R.I. 1979.Google Scholar
  16. [W2] Weinstein, A.: A fixed point theorem for positively curved manifolds, J. Math.Mech. 18 (1968), 149–153.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Xu Cheng
    • 1
  1. 1.IMPARio de JaneiroBrazil

Personalised recommendations