A Classification of Spherical Symmetric Cr Manifolds

Abstract

In this paper we get different characterizations of the spherical strictly pseudoconvex CR manifolds admitting a CR-symmetric Webster metric by means of the Tanaka-Webster connection and of the Riemannian curvature tensor. As a consequence we obtain the classification of the simply connected, spherical symmetric pseudo-Hermitian manifolds.2000 Mathematics subject classification: primary 53C15, 53C25, 53C35; secondary 32V05.

On the classification of contact metric ‐spaces via tangent hyperquadric bundles

“…then using Equation (2.6), by a standard computation (cf. also [9] in the Riemannian case), we obtain:…”

“…Since the symmetry at p is uniquely determined, it makes sense also to define locally symmetric pseudo‐Hermitian manifolds in a natural manner. Observe that, since the local symmetries are maps, for these manifolds the integrability condition is automatically satisfied (see ).…”

“…Finally, we recall that [, Theorem 3.2] a non Sasakian contact metric manifold satisfies the condition if and only if it is a locally symmetric pseudo‐Hermitian manifold.…”

“…Since locally then using Equation , by a standard computation (cf. also in the Riemannian case), we obtain: and hence becomes: It follows that , thus …”

“…Looking at contact metric manifolds as strongly pseudo‐convex (almost) manifolds, Dileo and Lotta showed that the ‐condition is equivalent to the local ‐symmetry with respect to the Webster metric g (see and Section ). In this context, another characterization was given by Boeckx and Cho and in terms of the parallelism of the Tanaka–Webster curvature and torsion .…”