Journal of Geometry

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

Riemannian manifolds with a parallel field of complex planes

  • Izu Vaisman
Article

Abstract

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.

Keywords

Riemannian Manifold Complex Plane Equal Length Betti Number General Presentation 

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Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

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

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