A new construction of homogeneous quaternionic manifolds and related geometric structures (Q2715731)

A quaternionic structure on a vector space \(E\) is a 3-dimensional subspace \(Q=\text{span}\{J_1,J_2,J_3\} \subset\text{End}(E)\) where \(J_1,J_2,J_3\) are complex structures on \(E\) satisfying \(J_1J_2=J_3\). If \(M\) is a smooth manifold, \(\dim M>4\), a field \(Q\), \(m\to Q_m\), \(Q_m\) a quaternionic structure on \(T_mM\), is a quaternionic structure on \(M\) if there exists a torsion free connection on \(TM\) preserving the rank 3 sub-bundle \(Q\subset \text{End}(TM)\).NEWLINENEWLINENEWLINEIn this paper the author studies various geometric features of homogeneous quaternionic manifolds constructed from \(\text{spin} (R^{p,q})\) equivariant linear maps \(\pi:\wedge^2 w\to R^{p,q}\) where \(W\) is a module over the even Clifford algebra \({\mathcal C}\ell^0 (R^{p,q})\) and \(p\geq 3\). One of the main consequences of the above construction is to obtain in a new and unified manner Alekseevsky spaces. Moreover, when \(p>3\) it is proved that the manifolds obtained do not admit any transitive action by a solvable Lie group, thus obtaining new families of quaternionic pseudo-Kähler manifolds. The author also studies the associated twistor spaces and other related bundles and shows that the above construction has a natural mirror in the category of supermanifolds.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].