Einstein and η -Einstein Sasakian submanifolds in spheres

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 η-Einstein cases when the codimension of the immersion is 4. Moreover, we exhibit infinite families of compact Sasakian η-Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show that, after possibly performing a 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.