## Journal of the Mathematical Society of Japan

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

### Foliated CR manifolds

Sorin DRAGOMIR and Seiki NISHIKAWA

#### Abstract

We study foliations on CR manifolds and show the following. (1) For a strictly pseudoconvex CR manifold $M$, the relationship between a foliation $\mathcal{F}$ on $M$ and its pullback ${\pi}^{*}\mathcal{F}$ on the total space $C\left(M\right)$ 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 $C\left(M\right)$, respectively. (2) With a tangentially CR foliation $\mathcal{F}$ on a nondegenerate CR manifold $M$, we associate the basic Kohn-Rossi cohomology of $(M,\mathcal{F})$ and prove that it gives the basis of the ${E}_{2}$-term of the spectral sequence naturally associated to $\mathcal{F}$. (3) For a strictly pseudoconvex domain $\Omega $ in a complex Euclidean space and a foliation $\mathcal{F}$ defined by the level sets of the defining function of $\Omega $ on a neighborhood $U$ of $\partial \Omega $, 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 $\mathcal{F}$. (4) For a foliation $\mathcal{F}$ 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

**Source**

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

**Dates**

First available in Project Euclid: 27 September 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1190905448

**Digital Object Identifier**

doi:10.2969/jmsj/1190905448

**Mathematical Reviews number (MathSciNet)**

MR2091416

**Zentralblatt MATH identifier**

1066.53059

**Subjects**

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

**Keywords**

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

#### Citation

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