Memoirs of the American Mathematical Society 2000; 63 pp; softcover Volume: 147 ISBN10: 0821821113 ISBN13: 9780821821114 List Price: US$44 Individual Members: US$26.40 Institutional Members: US$35.20 Order Code: MEMO/147/700
 Let \(V = {\mathbb R}^{p,q}\) be the pseudoEuclidean vector space of signature \((p,q)\), \(p\ge 3\) and \(W\) a module over the even Clifford algebra \(C\! \ell^0 (V)\). A homogeneous quaternionic manifold \((M,Q)\) is constructed for any \(\mathfrak{spin}(V)\)equivariant linear map \(\Pi : \wedge^2 W \rightarrow V\). If the skew symmetric vector valued bilinear form \(\Pi\) is nondegenerate then \((M,Q)\) is endowed with a canonical pseudoRiemannian metric \(g\) such that \((M,Q,g)\) is a homogeneous quaternionic pseudoKähler 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 pseudoKähler 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 \(Z \rightarrow M\) 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 \(Z \rightarrow \bar{Z}\) into a homogeneous complex manifold \(\bar{Z}\) of complex algebraic group is constructed. Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any \(\mathfrak{spin}(V)\)equivariant linear map \(\Pi : \vee^2 W \rightarrow V\) a homogeneous quaternionic supermanifold \((M,Q)\) is constructed and, moreover, a homogeneous quaternionic pseudoKähler supermanifold \((M,Q,g)\) if the symmetric vector valued bilinear form \(\Pi\) is nondegenerate. Readership Graduate students and research mathematicians interested in differential geometry. Table of Contents  Introduction
 Extended Poincaré algebras
 The homogeneous quaternionic manifold \((M,Q)\) associated to an extended Poincaré algebra
 Bundles associated to the quaternionic manifold \((M,Q)\)
 Homogeneous quaternionic supermanifolds associated to superextended Poincaré algebras
 Appendix. Supergeometry
 Bibliography
