CR Weyl geometry and connections of the canonical bundle (Q2494629)

Let \(M^{2n+1}\) be an odd-dimensional manifold. A CR structure on \(M\) consists of a codimension one distribution \(D\) and an almost complex structure \(J\) on \(D\) with some integrability property. A strictly pseudoconvex CR structure admits a natural contact form associated to it which also defines an Hermitian metric on \(D\). Then the structure is preserved by a unique connection with special torsion, called Tanaka-Webster connection. Since the change of the contact form leads to a conformal change of the Hermitian metric on \(D\) it is natural to consider a conformal invariant connection, called Weyl connection in analog with the Riemannian case. CR Weyl connections have been considered by the author and \textit{K. Sakamoto} [Tsukuba J. Math. 29, No. 2, 309--361 (2005; Zbl 1104.53042 reviewed below)] and \textit{L. David} [Ann. Global Anal. Geom. 26, No. 1, 59--72 (2004; Zbl 1054.32022)]. The paper under review considers the relation between an Einstein-like conditions on Tanaka-Webster and Weyl connections. It has some points in common with the above mentioned paper by L. David.