Rigidity of transversally conformally parallelizable foliations

This article is devoted to the study of foliations for which the notion of transverse direction has a global and intrinsic meaning. These foliations are said to be transversally conformally parallelizable (TCP). This work is a part of the analysis of bihamiltonian systems defined on odd dimensional manifolds. The dynamics of such a system is linked to the dynamics of a TCP foliation naturally associated with the bihamiltonian system. Firstly it will be proved that a connection is canonically attached to a TCP foliation. Thus the situation can be locally linearized. However, this connection is geodesically non-complete. Our study allows this difficulty to be overcome and a precise description of TCP foliations is obtained on closed manifolds in every dimension and codimension greater than 1. In particular, this leads to a classification of the TCP flows.