## Journal of the Mathematical Society of Japan

### Foliated CR manifolds

#### Abstract

We study foliations on CR manifolds and show the following. (1) For a strictly pseudoconvex CR manifold $M$, the relationship between a foliation $\mathscr{F}$ on $M$ and its pullback $\pi^{*}\mathscr{F}$ 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 $\mathscr{F}$ on a nondegenerate CR manifold $M$, we associate the basic Kohn-Rossi cohomology of $(M,\mathscr{F})$ and prove that it gives the basis of the $E_2$-term of the spectral sequence naturally associated to $\mathscr{F}$. (3) For a strictly pseudoconvex domain $\Omega$ in a complex Euclidean space and a foliation $\mathscr{F}$ 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 $\mathscr{F}$. (4) For a foliation $\mathscr{F}$ 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

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

Digital Object Identifier
doi:10.2969/jmsj/1190905448

Mathematical Reviews number (MathSciNet)
MR2091416

Zentralblatt MATH identifier
1066.53059

#### 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