On the classificarion of 3-dimensional spherical Sasakian manifolds

TL;DR

This paper classifies 3-dimensional spherical Sasakian manifolds by establishing parameter correspondences, describing the moduli space, and analyzing automorphism groups to distinguish homogeneous cases.

Contribution

It introduces a unified parameter framework for spherical hypersurfaces, describes their moduli space, and characterizes automorphism groups, advancing the understanding of Sasakian manifold classifications.

Findings

Established correspondence between different parameter sets for Reeb vector fields.

Described the moduli space of rigid spheres geometrically.

Identified automorphism groups and homogeneous cases among rigid spheres.

Abstract

In this article we consider spherical hypersurfaces in $\mathbb C^2$ 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 by Ezhov and Schmalz. Finally, we determine the Sasakian automorphism groups of the rigid spheres and detect the homogeneous Sasakian manifolds among them.