The Tanaka-Webster connection for almost S-manifolds and Cartan geometry

We prove that a CR-integrable almost S-manifold admits a canonical linear connection, which is a natural generalization of the Tanaka– Webster connection of a pseudo-hermitian structure on a strongly pseudo- convex CR manifold of hypersurface type. Hence a CR-integrable almost S-structure on a manifold is canonically interpreted as a reductive Cartan geometry, which is torsion free if and only if the almost S-structure is normal. Contrary to the CR-codimension one case, we exhibit examples of non normal almost S-manifolds with higher CR codimension, whose Tanaka–Webster curvature vanishes