Sasakian immersions into the sphere

TL;DR

This paper classifies compact Sasakian manifolds that can be immersed into spheres, especially in Einstein and η-Einstein cases, and explores conditions for such immersions involving homothetic deformations and fundamental groups.

Contribution

It provides a complete classification of Sasakian immersions into spheres in specific Einstein cases and identifies infinite families that cannot be immersed, advancing understanding of Sasakian geometry.

Findings

Complete classification in Einstein and η-Einstein cases for codimension 4.

Existence of infinite families not immersible into spheres.

Characterization of homogeneous Sasakian manifolds that admit immersions after deformation.

Abstract

The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $\eta$-Einstein cases when the codimension of the immersion is $4$. Moreover, we exhibit infinite families of compact Sasakian $\eta$--Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show that, after possibly performing a ${\mathcal {D}}$-homothetic deformation, a homogeneous Sasakian manifold can be Sasakian immersed into some odd-dimensional sphere if and only if $S$ is regular and either $S$ is simply--connected or its fundamental group is finite cyclic.