Rigid spheres and homogeneous Sasakian manifolds

Sasakian manifolds can be defined as CR manifolds with a fixed translational symmetry transversal to the CR distribution. Locally, a Sasakian manifold of dimension $2n+1$ can be embedded into $\mathbb C^{n+1}$ as a real hypersurface with defining equation $\mathrm{Im} w=f(z)$, which does not depend on $\mathrm{Re} w$. Such hypersurfaces have been coined "rigid’’. N. Stanton has developed a version of the Chern-Moser normal form that takes into account rigidity. Rigidity can also be considered as a weaker structure than a Sasakian structure, by fixing a translational symmetry only up to scale.
Stanton’s rigid normal form is very useful in the study of Sasakian and rigid structures. We demonstrate this in relation to homogeneous 3-dimensional Sasakian and rigid manifolds.
This is joint work with V. Ezhov and D. Sykes.