In this paper we study positive solutions to the CR Yamabe equation on groups of Heisenberg type. For the subclass of groups of Iwasawa type, we characterize those solutions that have partial symmetry, that is, those that are invariant with respect to the action of the orthogonal group in the first layer of the Lie algebra.The exponential map exp : g → G is an analytic diffeomorphism. It induces a group of dilations on G via the formula δ λ (g) = exp • λ • exp −1 (g), g ∈ G.
A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan-Chern-Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in C n+1 is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.
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