On Symmetric Cauchy–Riemann Manifolds

A Classification of Spherical Symmetric Cr Manifolds

“…To end this section, we shall prove that for Sasakian manifolds, local symmetry in the sense of [19] is actually equivalent to a similar concept in the literature, namely locally ϕ-symmetric contact metric structure (see [3,6]). The latter is defined by the requirement that the characteristic reflections, that is, the reflections with respect to the integral curves of ξ , be local isometries.…”

“…We shall also say that g is a CR-symmetric Hermitian metric on (M, H M, J ). Since the symmetry at x is uniquely determined (see [19,Theorem 3.3]) it makes sense also to define Hermitian locally CR-symmetric almost CR spaces in a natural manner. Observe that, since the symmetries are CR maps, for this class of almost CR spaces the integrability condition (2.2) is automatically satisfied.…”

“…It is proved in [19] that a Hermitian symmetric CR space M is CR-homogeneous: in fact the subgroup of the automorphism group Aut C R (M) generated by the symmetries acts transitively. In particular, every Hermitian symmetric CR space M is a real analytic CR manifold.…”

“…In this paper we adopt a geometric point of view in studying spherical CR manifolds, concentrating our attention on CR-symmetric Webster metrics g η . For the general notion of a symmetric Hermitian metric on a CR manifold we refer to [19] (see also Section 3). Here we recall that a Webster metric g η is CR-symmetric if for each point x ∈ M there exists a CR-isometry σ : M → M with σ (x) = x and (dσ ) x |H x M = −Id.…”

A new hypoelliptic operator on almost CR manifolds

“…However, as explained in Section 1, there is a handful of interesting examples of CR manifolds which do admit a real structure. Among these are real hypersurfaces of C n+1 that are rigid in the sense of [2], nilpotent Lie groups and CR nilmanifolds, some CR symmetric spaces in the sense of [12], and contact manifolds equipped with a Legendrian subbundle, including circle bundles associated to the geometric quantization of symplectic manifolds.…”