A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures

Let V = R be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cl0(V ). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V )-equivariant linear map Π : ∧2W → V . If the skew symmetric vector valued bilinear form Π is nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q, g) is a homogeneous quaternionic pseudo-Kähler manifold. If the metric g is positive definite, i.e. a Riemannian metric, then the quaternionic Kähler 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 Kähler 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 quaternionic pseudo-Kähler 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 Z → M and the canonical SO(3)-principal bundle S → 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 D of complex codimension one, which is a complex contact structure if and only if Π is nondegenerate. Moreover, an equivariant Supported by SFB 256 (Bonn University). e-mail: [email protected]