Lie bialgebroids of generalized CRF-manifolds

Abstract

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 ξ ± η , where ξ is a vector field and η is a 1-form on M with η ( ξ ) = 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 ξ ± η , 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.