We define a generalized pseudohermitian structure on an almost CR manifold (M,HM,J) as a pair (h,P), where h is a positive definite fiber metric h on HM compatible with J, and P:TM-> TM is a smooth projector such that Im(P)=HM. We show that to each generalized pseudohermitian structure one can associate a canonical linear connection on the holomorphic bundle HM which is invariant under equivalence. This fact allows us to solve the equivalence problem in the case where HM is a kind 2 distribution. We study the curvature of the canonical connection, especially for the classes of standard homogeneous CR manifolds and 3-Sasakian manifolds. The basic formulas for isopseudohermitian immersions are also obtained in the attempt to enlarge the theory of pseudohermitian immersions between strongly pseudoconvex pseudohermitian manifolds of hypersurface type.

We study the Riemannian geometry of contact manifolds with respect to a fixed admissible metric, making the Reeb vector field unitary and orthogonal to the contact distribution, under the assumption that the Levi-Tanaka form is parallel with respect to a canonical connection with torsion.

We prove that there exist no simply connected homogeneous contact metric manifolds having nonpositive sectional curvature.

We characterize and study Riemannian almost CR manifolds admitting characteristic connections, i.e. metric connections with totally skew-symmetric torsion, parallelizing the almost CR structure. Natural constructions are provided of new non trivial 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).

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).

We describe some new examples of nilmanifolds admitting an Einstein with skew torsion invariant Riemannian metric. These are affine CR quadrics, whose CR structure is preserved by the characteristic connection.