Any Sasakian structure is approximated by embeddings into spheres

TL;DR

This paper proves that any Sasakian structure on a closed manifold can be closely approximated by structures derived from CR embeddings into weighted spheres, extending previous approximation results to higher regularity.

Contribution

It introduces a method to approximate any Sasakian structure with CR embeddings into spheres and strengthens existing orbifold Kähler form approximations to higher regularity.

Findings

Any Sasakian structure can be approximated by CR embeddings into spheres.

The approximation of orbifold Kähler forms is extended to $C^{q}$-norm.

Provides a new link between Sasakian geometry and CR embeddings.

Abstract

We show that, for any given $q\geq 0$, any Sasakian structure on a closed manifold $M$ is approximated in the $C^{q}$-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold K\"ahler form by projectively induced ones given by Ross and Thomas in [21] in the $C^0$-norm to a $C^{q}$-approximation.