A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natural way. For the case of the odd dimensional spheres S 2n+1 viewed as homogeneous spaces of the special unitary groups, the classical Nomizu's Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the spheres S 2n+1 into homogeneous Riemann-Cartan spaces. The expressions of such connections as covariant derivatives are given by means of several invariant tensors: the ones of the usual Sasakian structure of the sphere; an invariant 3-differential form coming from a 3-Sasakian structure on S 7 ; and the involved ones in the almost contact metric structure of S 5 provided by its natural embedding into the nearly Kähler manifold S 6 . Furthermore, the invariant connections sharing geodesics with the Levi-Civita one have also been completely described. Finally, S 3 and S 7 are characterized as the unique odd-dimensional spheres which admit nontrivial invariant connections satisfying an Einstein-type equation.2010 Mathematics Subject Classification. Primary 53C30, 53C05; Secondary 53C25, 53C20.
The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections which are Einstein with skew-torsion up to S 3 . For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemaniann Berger spheres, not Einstein, which admit invariant Einstein with skew-torsion affine connections.2010 Mathematics Subject Classification. Primary 53C05; Secondary 53C30, 53C25, 53C20.
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