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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 6, pp 2195–2205 | Cite as

Einstein and \(\eta \)-Einstein Sasakian submanifolds in spheres

  • Beniamino Cappelletti-Montano
  • Andrea LoiEmail author
Article
  • 80 Downloads

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.

Keywords

Sasakian Sasaki–Einstein \(\eta \)-Einstein Sasakian immersion Kähler manifolds Kähler immersions 

Mathematics Subject Classification

53C25 53C55 

Notes

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly

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