A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures

Let $V = mathbb Rp,q$ be the pseudo-Euclidean vector space of signature $(p,q)$, $pge 3$ and $W$ a module over the even Clifford algebra $C!ell0 (V)$. A homogeneous quaternionic manifold $(M,Q)$ is constructed for any $mathfrak spin (V)$-equivariant linear map $ wedge2 Wrightarrow V$. If the skew symmetric vector valued bilinear form $Pi$ is nondegenerate then $(M,Q)$ is endowed with a canonical pseudo-Riemannian metric $g$ such that $(M,Q,g)$ is a homogeneous quatemionic pseudo-Kahler manifold. If the metric $g$ is positive definite, i.e. a Riemannian metric, then the quaternionic Kahler manifold $(M,Q,g)$ is shown to admit a simply transitive solvable group of automorphisms.In this special case ($p=3$) we recover all the known homogeneous quaternionic Kahler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If $p>3$ then $M$ does not admit any transitive action of a solvable Lie group and we obtain new families of quatermionic pseudo-Kahler manifolds. Then it is shown that for $q = 0$ the noncompact quaternionic manifold $(M,Q)$ can be endowed with a Riemannian metric $h$ such that $(M,Q,h)$ is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if $p>3$. The twistor bundle $Zrightarrow M$ and the canonical $mathrm SO (3)$-principal bundle $S rightarrow M$ associated to the quaternionic manifold $(M,Q)$ are shown to be homogeneous under the automorphism group of the base.More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution $mathcal D$ of complex codimension one, which is a complex contact structure if and only if $Pi$ is nondegenerate. Moreover, an equivariant open holomorphic immersion $ZrightarrowbarZ$ into a homogeneous complex manifold $barZ$ of complex algebraic group is constructed. Finally, the construction is shown to have a natural mirror in the category of supermanifolds