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 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.