On the Tanno connection and the Chern-Moser connection, in almost CR-geometry

Abstract

We show that one can construct a Cartan connection on the Cartan principal bundle over a contact Riemannian manifold, on which the associated complex structure is not assumed to be integrable, according to Cartan-Chern-Moser-Le's construction but with the use of the Tanno connection (instead of the Tanaka-Webster connection in the integrable case). Then we prove that it is normal in the sense of Tanaka if and only if the complex structure is integrable. By Le this has been shown to hold true in the case the dimension of the manifold is three.