Journal of Geometry

, Volume 17, Issue 1, pp 174–192 | Cite as

Riemannian manifolds with a parallel field of complex planes

  • Izu Vaisman


The purpose of this paper is to discuss Riemannian manifolds which admit a parallel field of complex planes, consisting of vectors of the form\(a + \sqrt { - 1} b\), where a,b are real orthogonal vectors of equal length. Using the Nirenberg Frobenius Theorem [12], it follows that these are reducible Riemannian manifolds, whose metric is locally a sum of a Kähler and of a Riemann metric, and we are calling thempartially Kähler manifolds.

After a general presentation of these manifolds (including a general presentation of the complex integrable plane fields) we are discussing harmonic forms, Betti numbers, and Dolbeault cohomology. This discussion is based on a theorem of Chern [4], and it provides generalizations of the results of Goldberg [9], as well as some other new results.


Riemannian Manifold Complex Plane Equal Length Betti Number General Presentation 
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  1. 1.
    C.M. de Barros, Variétés preseque hor-complexes, C.R. Acad. Sci. Paris 260 (1965), 1543–46.Google Scholar
  2. 2.
    D.E. Blair, Geometry of manifolds with structural group U(n)xO(s), J. Differential Geometry 4 (1970), 155–168.Google Scholar
  3. 3.
    R. Bott, Lectures on characteristic classes and foliations, Lecture Notes in Math., 279, Springer-Verlag, New New York, 1972, 1–94.Google Scholar
  4. 4.
    S.S. Chern, On a generalization of Kähler geometry, Algebraic Geometry and Topology, A Symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, 1957, 103–121.Google Scholar
  5. 5.
    L. A. Cordero and P.M. Gadea, Exotic characteristic classes and subfoliations, Ann. Inst. Fourier (Grenoble) 26 (1976), 225–237.Google Scholar
  6. 6.
    T. Duchamp and M. Kalka, Deformation theory and stability for holomorphic foliations, Geometry and Diff. Geometry, Proceedings, Haifa, Israel, 1979, Lect. Notes in Math 792, Springer-Verlag, Heidelberg, 1980, pp. 235–246.Google Scholar
  7. 7.
    H.R. Fischer and F.L. Williams, Complex-foliated structures I. Cohomology of the Dolbeault-Konstant complexes, Transactions American Math. Soc., 252 (1979), 163–195.Google Scholar
  8. 8.
    S.I. Goldberg, Curvature and Homology, Academic Press, New York, 1962.Google Scholar
  9. 9.
    S.I. Goldberg, A generalization of Kähler geometry, J. Differential Geometry, 6 (1972), 343–355.Google Scholar
  10. 10.
    S. Kashiwabara, The structure of a Riemannian manifold admitting a parallel field of tangent vector subspaces, Tôhoku Math. J. 12 (1960), 102–119.Google Scholar
  11. 11.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I. II, Interscience Publ., New York, 1963, 1969.Google Scholar
  12. 12.
    L. Nirenberg, A complex Frobenius theorem, Seminar on Analytic Functions, Inst. for Advanced Studies, Princeton, I (1957), 172–179.Google Scholar
  13. 13.
    J.H. Rawnsley, On the cohomology groups of a polarization and diagonal quantization, Transactions American Math. Soc. 230 (1977), 235–255.Google Scholar
  14. 14.
    B.L. Reinhart, Harmonic integrals on almost product manifolds, Transactions American Math. Soc. 88 (1958), 243–276.Google Scholar
  15. 15.
    E.H. Spanier, Algebraic topology, McGraw-Hill, Comp., New York, 1966.Google Scholar
  16. 16.
    I. Vaisman, From the geometry of Hermitian foliate manifolds, Bull. Math, de la Soc. Math. de la R. S. de Roumanie, 17 (1973), 71–100.Google Scholar
  17. 17.
    I. Vaisman, A coordinatewise formulation of geometric quantization, Ann. Inst. H. Poincaré 31 (1979), 5–24.Google Scholar
  18. 18.
    A. Weil, Introduction à l'étude des variétés Kähleriennes, Hermann, Paris, 1958.Google Scholar
  19. 19.
    K. Yano, On a structure defined by a tensor field f of type (1,1) satisfying f3+f = 0, Tensor 14 (1963), 99–109.Google Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • Izu Vaisman
    • 1
  1. 1.Department of MathematicsUniversity of HaifaIsrael

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