Homogeneous non-degenerate 3-\((\alpha,\delta)\)-Sasaki manifolds and submersions over quaternionic Kähler spaces (Q2033064)

The first two authors of this article have recently introduced a class of almost 3-contact metric manifolds called 3-\((\alpha,\delta)\)-Sasaki manifolds. Here, \(\alpha\neq 0\) and \(\delta\) are real constants; in the case \(\alpha=\delta\) one recovers the previously studied notion of a 3-\(\alpha\)-Sasaki structure, which in turn reduces to a 3-Sasaki structure for \(\alpha=1\). This work concerns the fundamental properties of these 3-\((\alpha,\delta)\)-Sasaki manifolds. It is known that each 3-\((\alpha,\delta)\)-Sasaki manifold of dimension \(4n+3\) admits a unique metric connection with skew torsion which is compatible with the full structure in a suitable sense, called the canonical connection. The authors show that the general properties of the 3-\((\alpha,\delta)\)-Sasaki structure and the canonical connection imply that any 3-\((\alpha,\delta)\)-Sasaki manifold locally has the structure of a Riemannian submersion with totally geodesic fibers, the tangent spaces to which are spanned by the triple of Reeb vector fields. The main theorems proven in the first half of the paper then assert that the \(4n\)-dimensional base of this Riemannian submersion carries the structure of a quaternionic Kähler manifold, and that its scalar curvature equals \(16n(n+2)\alpha\delta\). The second half of the paper is concerned with the construction of interesting examples. Recall that quaternionic Kähler manifolds with \(\operatorname{scal}=0\) (which are in fact hyper-Kähler), \(\operatorname{scal}>0\) and \(\operatorname{scal}<0\) each have drastically different properties; this motivates one to distinguish the cases \(\delta=0\), \(\alpha\delta>0\) and \(\alpha\delta<0\), which are called degenerate, positive and negative 3-\((\alpha,\delta)\)-Sasaki manifolds, respectively. Using Lie-theoretic methods, the authors discuss the construction of homogeneous, non-degenerate examples. The corresponding quaternionic Kähler manifold is then homogeneous of non-zero scalar curvature, and in the positive case this implies that it must be a compact quaternionic Kähler symmetric space. The authors give an explicit construction of the homogeneous 3-\((\alpha,\delta)\)-Sasaki manifolds corresponding to these so-called (compact) Wolf spaces, as well as their non-compact duals. The negative examples coming from the non-compact duals do not exhaust the class of negative, homogeneous 3-\((\alpha,\delta)\)-Sasaki manifolds: Further examples arise from the additional homogeneous quaternionic Kähler manifolds of negative scalar curvature discovered by Alekseevsky. A precise description of these 3-\((\alpha,\delta)\)-Sasaki manifolds is given as well, after which the authors discuss a number of low-dimensional examples in more detail. In the final section of the article, the authors describe the canonical connection of a general homogeneous 3-\((\alpha,\delta)\)-Sasaki manifold in Lie-algebraic terms. They subsequently specialize to the case where the quaternionic Kähler base is symmetric, where they obtain a simplified description.