Einstein and $$\eta $$-Einstein Sasakian submanifolds in spheres

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 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.

Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

“…We believe that these results can be generalized to arbitrary codimension. Such a generalization of Corollary 3 and Corollary 4 is related to an analogous conjecture on Sasakian immersions of η-Einstein manifolds, see [8] and referecences therein.…”

“…the standard sphere S 2N +1 . Very little is known in this setting, compared to the non-compact case, even in the Sasakian manifold is assumed to be eta-Einstein, see [8]. For this reason we state a general result and several corollaries under additional or different hypotheses.…”

“…In fact, Ornea and Verbitsky [15] proved that any compact Sasakian manifold admits a CR embedding in a Sasakian manifold diffeomorphic to a sphere. The case of η-Einstein manifolds admitting a Sasakian immersion has been, in some instances only partially, solved in [1,8,12]. On the other hand, this question has not been addressed for Sasaki-Ricci solitons.…”

“…Firstly, let us consider the case where S is compact and the codimension of the immersion is arbitrary, cf. Theorems 1 and 3 in [8].…”

“…Lastly we turn our attention to the case of small codimension to prove the two following corollaries, cf. part (i) of the main theorem of [12] and [8,Theorem 2].…”