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Affine connections on 3-Sasakian and manifolds

  • Cristina Draper
  • Miguel OrtegaEmail author
  • Francisco J. Palomo
Article
  • 54 Downloads

Abstract

The space of invariant affine connections on every 3-Sasakian homogeneous manifold of dimension at least seven is described. In particular, the subspace of invariant affine metric connections and the subclass with skew torsion are also determined. To this aim, an explicit construction of all 3-Sasakian homogeneous manifolds is exhibited. It is shown that the 3-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew torsion invariant affine connections are those of dimension seven, that is, \({\mathbb {S}}^7\), \({\mathbb {R}}P^7\) and the Aloff–Wallach space \({\mathfrak {W}}^{7}_{1,1}\). On \({\mathbb {S}}^7\) and \({\mathbb {R}}P^7\), the set of such connections is bijective to two copies of the conformal linear transformation group of the Euclidean space, while it is strictly bigger on \({\mathfrak {W}}^{7}_{1,1}\). The set of invariant connections with skew torsion whose Ricci tensor satisfies that its eigenspaces are the canonical vertical and horizontal distributions, is fully described on 3-Sasakian homogeneous manifolds. An affine connection satisfying these conditions is distinguished, by parallelizing all the Reeb vector fields associated with the 3-Sasakian structure, which is also Einstein with skew torsion on the 7-dimensional examples. The invariant metric affine connections on 3-Sasakian homogeneous manifolds with parallel skew torsion have been found. Finally, some results have been adapted to the non-homogeneous setting.

Keywords

3-Sasakian homogeneous manifolds Invariant affine connections Riemann–Cartan manifolds Einstein with skew torsion connections Ricci tensor Parallel skew torsion Compact simple Lie algebra 

Mathematics Subject Classification

Primary 53C25 53C30 53B05 Secondary 53C35 17B20 17B25 

Notes

Acknowledgements

The authors would like to thank the referee for the deep reading of this manuscript, for his/her valuable suggestions to include Sect. 5.6 on parallel torsion and for several comments on the bibliography.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, Escuela de Ingenierías IndustrialesUniversidad de MálagaMálagaSpain
  2. 2.Departamento de Geometría y Topología, Facultad de Ciencias, Instituto de Matemáticas IEMathUGRUniversidad de GranadaGranadaSpain

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