On the Classification of Spherical Rigid CR Manifolds and Sasakian Manifolds in C2

Abstract

<p>We consider spherical hypersurfaces in C² with a fixed Reeb vector field as 3-dimensional Sasakian manifolds. We establish the correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, the parameters used in Stanton’s description of rigid spheres, and the parameters arising from the rigid normal forms. We also geometrically describe the moduli space for rigid spheres, and provide geometric distinction between Stanton’s hypersurfaces and those found in [17]. We determine the Sasakian automorphism groups of the rigid spheres, detecting the homogeneous Sasakian manifolds amongst them, and we determine the Sasakian automorphisms of the CR manifolds arising in E. Cartan’s classical list of homogeneous CR hypersur- ´ faces. Furthermore, we relax the condition on the Reeb vector field to allow preservation up to a nonzero dilation, called homothetic Sasakian preservation. Finally, we determine the homogeneous Sasakian manifolds with respect to the homothetic Sasakian preservation.</p>