Lie bialgebroids of generalized CRF-manifolds

The notion of a generalized CRF-structure on a smooth manifold was recently introduced and studied by Vaisman (2008) [6]. An important class of generalized CRF-structures on an odd dimensional manifold M consists of CRF-structures having complementary frames of the form xi +/- eta, where xi is a vector field and eta is a 1-form on M with eta(xi) = 1. It turns out that these kinds of CRF-structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [5]. More precisely, we show that to any CRF-structures with complementary frames of the form xi +/- eta there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a Cauchy-Riemann structure on a manifold. (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.