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Affine connections on 3-Sasakian and manifolds

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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.

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Change history

  • 27 July 2019

    In the original publication of this article, the title was incorrectly published as “<Emphasis Type="Bold">Affine connections on 3-Sasakian and manifolds</Emphasis>”. The correct title should be “<Emphasis Type="Bold">Affine connections on 3-Sasakian homogeneous manifolds</Emphasis>”.

Notes

  1. Our convention is \(d\eta (X,Y)=X(\eta (Y))-Y(\eta (X))-\eta ([X,Y])\).

  2. Recall that there is exactly one irreducible \(\mathfrak {sl}(2,{\mathbb {C}})\)-module of each dimension \(n+1\), which is frequently denoted by V(n). It coincides with \(V(n)\cong S^n(V(\lambda _1))\). In particular \(V(1)\cong V(\lambda _1)\cong {\mathbb {C}}^2\) is given by the columns.

  3. Our convention for the exterior product of a p-form \(\omega _1\) and a q-form \(\omega _2\) is the following

    $$\begin{aligned} \omega _1\wedge \omega _2(X_1,\dots ,X_{p+q})= \frac{1}{p!q!}{\sum }_{\sigma \in S_{p+q}}(-1)^{[\sigma ]} \omega _1(X_{\sigma (1)},\dots ,X_{\sigma (p)})\omega _1(X_{\sigma (p+1)},\dots ,X_{\sigma (p+q)}). \end{aligned}$$

  4. Be aware that the expression of \(\omega _{_{\nabla }}\) on the preliminar version arXiv:1503.08401v1 have a wrong coefficient \(\frac{1}{2}\), corrected in the version [22].

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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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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Universidad de Málaga, 29071, Málaga, Spain

    Cristina Draper & Francisco J. Palomo

  2. Departamento de Geometría y Topología, Facultad de Ciencias, Instituto de Matemáticas IEMathUGR, Universidad de Granada, 18071, Granada, Spain

    Miguel Ortega

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  1. Cristina Draper
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Correspondence to Miguel Ortega.

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C. Draper is partially supported by the Spanish Ministery of Economy and Competitiveness (MEC), and European Region Development Fund (ERDF), project MTM2016-76327-C3-1-P and by the Junta de Andaluía Grant FQM-336. F. J. Palomo is partially supported by the Spanish MEC, and ERDF, project MTM2016-78807-C2-2-P. M. Ortega is partially supported by the Spanish MEC, and ERDF, project MTM2016-78807-C2-1-P. Also, M. Ortega and F. J. Palomo are partially supported by the Junta de Andalucía Grant FQM-324.

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Draper, C., Ortega, M. & Palomo, F.J. Affine connections on 3-Sasakian and manifolds. Math. Z. 294, 817–868 (2020). https://doi.org/10.1007/s00209-019-02304-x

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