Foliated CR manifolds

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 $C(M)$ 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 $\mathcal F$ 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 $M$.