Quaternionic contact \(4n+3\)-manifolds and their \(4n\)-quotients
The author studies quaternionic contact structure (``qc-structure'' for short) on \((4n+3)\)-manifolds to construct quaternionic Hermitian \(4n\)-manifolds as their quotient. The qc-structure was first introduced by \textit{O. Biquard} [Métriques d'Einstein asymptotiquement symétriques. Paris: Société Mathématique de France (2000; Zbl 0967.53030)], together with a canonical connection (now known as ``Biquard connection''). A qc-structure on a \((4n+3)\)-manifold \(X\) is a distribution \(D\subset TX\) of codimension 3, such that \(D\) admits a \(n\)-dimensional quaternionic structure \(Q\). In a previous work of the author with \textit{D. Alekseevsky} [Ann. Mat. Pura Appl. (4) 187, No. 3, 487--529 (2008; Zbl 1223.53054)], a special case of qc-structures, called ``quaternionic CR-structure'' was studied, which corresponds to qc-Einstein manifolds with nonzero qc-scalar curvatures. In this paper, the author studies the complementing vanishing qc-scalar curvature case, which has an equivalent description as ``strict qc-structure''. In the first part of the paper, the author construct a family of simply connected strict qc solvable Lie groups \(\mathcal{M}_{k,l}\), starting with the standard quaternionic Heisenberg nilpotent Lie group \(\mathcal{M}\). These are then characterized as the only contractible unimodular strict qc-groups. Then, both compact and non-compact uniformizable strict qc-manifolds are classified modeling on \(\mathcal{M}\). In the second part, the author uses the quotients of a conformal deformation of the standard qc-structure on \(\mathcal{M}\) to construct a family of quaternionic Hermitian metrics on a domain of the standard quaternion space, one of which is a Bochner-flat Kähler metric.
scientific article; zbMATH DE number 7358590
scientific article