Journal of the Mathematical Society of Japan

Foliated CR manifolds


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We study foliations on CR manifolds and show the following. (1) For a strictly pseudoconvex CR manifold M, the relationship between a foliation F on M and its pullback π*F on the total space C(M) of the canonical circle bundle of M is given, with emphasis on their interrelation with the Webster metric on M and the Fefferman metric on C(M), respectively. (2) With a tangentially CR foliation F on a nondegenerate CR manifold M, we associate the basic Kohn-Rossi cohomology of (M,F) and prove that it gives the basis of the E2-term of the spectral sequence naturally associated to F. (3) For a strictly pseudoconvex domain Ω in a complex Euclidean space and a foliation F defined by the level sets of the defining function of Ω on a neighborhood U of Ω, we give a new axiomatic description of the Graham-Lee connection, a linear connection on U which induces the Tanaka-Webster connection on each leaf of F. (4) For a foliation F on a nondegenerate CR manifold M, we build a pseudohermitian analogue to the theory of the second fundamental form of a foliation on a Riemannian manifold, and apply it to the flows obtained by integrating infinitesimal pseudohermitian transformations on M.

Article information

J. Math. Soc. Japan, Volume 56, Number 4 (2004), 1031-1068.

First available in Project Euclid: 27 September 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]
Secondary: 32V05: CR structures, CR operators, and generalizations 32V40: Real submanifolds in complex manifolds 53C50: Lorentz manifolds, manifolds with indefinite metrics

tangentially CR foliation Fefferman metric Tanaka-Webster connection Graham-Lee connection basic Kohn-Rossi cohomology infinitesimal pseudohermitian transformation


DRAGOMIR, Sorin; NISHIKAWA, Seiki. Foliated CR manifolds. J. Math. Soc. Japan 56 (2004), no. 4, 1031--1068. doi:10.2969/jmsj/1190905448.

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