Correction to: Affine connections on 3-Sasakian homogeneous manifolds

Abstract

The space of invariant affine connections on every 3-Sasakian homogeneous manifold of dimension at least 7 is described. In particular, the subspaces 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 7, that is, S 7 , RP 7 and the Aloff-Wallach space W 7 1,1 . On S 7 and RP 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 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 and invariant affine connections and Riemann-Cartan manifolds and Einstein with skew-torsion connections and Ricci tensor and parallel skew-torsion and compact simple Lie algebra.

Homogeneous non-degenerate 3-(α,δ)-Sasaki manifolds and submersions over quaternionic Kähler spaces

“…For clarity we will denote the compact top Lie group by G and the non-compact one by G * . (d) By [12] any 3-Sasaki data gives rise to a homogeneous 3-Sasaki manifold. They were completely determined in [9] by the fact that they are fiber-bundles over the quaternionic Kähler base space G∕G 0 .…”

“…Section 3 is therefore devoted to a hands on construction of homogeneous 3-( , )-Sasaki spaces over all known homogeneous qK manifolds. This yields a construction over symmetric Wolf spaces, deforming the description given in [12] (see also [6,Theorem 4]), and by similar means their non-compact duals. Additionally, homogeneous 3-( , )-Sasaki manifolds over Alekseevsky spaces are constructed using a description of the latter given by Cortés [10].…”

Homogeneous Einstein manifolds based on symplectic triple systems

“…This implies that the standard enveloping algebra g(T ) is not only simple but central simple, i.e., g(T ) C is a simple complex Lie algebra. This will allow us to use the computations in [15], [13], since, for a 3-Sasakian homogeneous manifolds M = G/H, there exists a complex simple symplectic triple system W such that (Lie(G) C , Lie(H) C ) ∼ = (g(W ), inder(W )).…”

“…This note arises from the Second International Workshop on Nonassociative algebras held in Porto, May 2019. My talk there dealt with some results on 3-Sasakian manifolds [15] and how their close relationship with complex symplectic triple systems could bolster the study of some concrete questions related to curvature and holonomy [13]. The fact that, for a real symplectic triple system, the standard enveloping algebra is never a compact Lie algebra, made me wonder how the geometry of the related manifold could be.…”

“…for κ the Killing form of g. Observe that the choice of this metric is inspired in the Einstein metric in any 3-Sasakian homogeneous manifold [15,Theorem 4.3ii)]. Our knowledge of the Killing form shows, as in [13,Eq.…”