Affine Connections on 3-Sasakian Homogeneous Manifolds

The family of all $3$-Sasakian homogeneous manifolds is completely described. We characterize the 3-sphere $\mathbb{S}^3=\mathrm{Sp}(1)$, the 7-sphere $\mathbb{S}^7=\mathrm{Sp}(2)/ \mathrm{Sp(1)}$ and the Aloff-Wallach space $\mathfrak{W}^{7}_{1,1}=\mathrm{SU}(3)/ \mathrm{U}(1)$ as the unique {simply connected} examples of $3$-Sasakian homogeneous manifolds which admit non-trivial Einstein with skew-torsion invariant affine connections. For the non-necessarily homogeneous setting, in dimension $7$, we conclude that the set of non-trivial Einstein with skew-torsion affine connections contains two copies of the conformal linear transformation group of the Euclidean space, which exhausts the set in the sphere $\mathbb{S}^7$ under the hypothesis of invariance, while it is strictly bigger for $\mathfrak{W}^{7}_{1,1}$. In addition, on 3-Sasakian homogeneous manifolds of arbitrary dimension $\geq 7$, we fully describe the set of invariant connections with skew-torsion such that its Ricci tensor is (i) symmetric, and (ii) a multiple of the metric, with different factors, on the canonical horizontal and vertical distributions. An affine connection satisfying these conditions is distinguished, characterized by parallelizing all the characteristic vector fields associated with the $3$-Sasakian structure. This connection is Einstein with skew-torsion when the dimension is $7$.