Riemannian almost CR manifolds with torsion

Abstract

We characterize and study Riemannian almost CR manifolds admitting characteristic connections, that is, metric connections with totally skew-symmetric torsion parallelizing the almost CR structure. Natural constructions are provided of new nontrivial examples. We study the influence of the curvature of the metric on the underlying almost CR structure. A global classification is obtained under flatness assumption of a characteristic connection, provided that the fundamental $2$-form of the structure is closed (quasi Sasakian condition).