Addendum to 'The Geometry of Hyperbolic and Elliptic CR-Manifolds of Codimension Two'

Abstract

The aim of this article is to show how the individual harmonic components of the torsion of the canonical Cartan connection of embedded hyperbolic and elliptic CRmanifolds at a given point can be read off from the third order terms of the defining equation given in normal form. The general theory ensures that the vanishing of each of these one-dimensional components implies striking geometric consequences and we link each of them to an easily computable coefficient in the normal form. This allows to correct a mistake in [SS00] where it was claimed that four torsion components out of six vanish automatically for embedded CR-manifolds. The failure in that article appears already in Lemma 1.1 where the second order osculation was not dealt with carefully enough. At the same time, the rest of [SS00] is essentially worked out for abstract CR–structures and so the validity of the procedures and results has not been effected in general. In what follows, we use the terminology and notation of [SS00] without further comments.