We study D-homothetic deformations of almost alpha-Kenmotsu structures. We characterize almost contact metric manifolds which are CR-integrable almost alpha-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under D-homothetic deformations. If the canonical connection associated to the structure (phi, xi, eta, g) has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator h' defined by 2 alpha h' = (L xi phi) o phi. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost alpha-Kenmotsu structure. In the case of almost a-Kenmotsu (k, mu)'-spaces, this classification gives rise to a scalar invariant depending on the real numbers K and alpha.

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 analyze the Riemannian geometry of almost alpha-Kenmotsu manifolds, focusing on local symmetries and on some vanishing conditions for the Riemannian curvature. If the characteristic vector field of an almost alpha-Kenmotsu structure belongs to the so-called (kappa,mu)'-nullity distribution, $\kappa < -\alpha^2$, then the Riemannian curvature is completely determined. These manifolds provide a special case of a wider class of almost alpha-Kenmotsu manifolds, for which an operator h' associated to the structure is eta-parallel and has constant eigenvalues. All these manifolds are locally warped products. Finally, we give a local classification of almost alpha-Kenmotsu manifolds, up to D-homothetic deformations. Under suitable conditions, they are locally isomorphic to Lie groups.

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.