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A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures
Vicente Cortés, University of Bonn, Germany
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Memoirs of the American Mathematical Society
2000; 63 pp; softcover
Volume: 147
ISBN-10: 0-8218-2111-3
ISBN-13: 978-0-8218-2111-4
List Price: US$47 Individual Members: US$28.20
Institutional Members: US\$37.60
Order Code: MEMO/147/700

Let $$V = {\mathbb R}^{p,q}$$ be the pseudo-Euclidean 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 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 $$Z \rightarrow M$$ 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 pseudo-Kähler supermanifold $$(M,Q,g)$$ if the symmetric vector valued bilinear form $$\Pi$$ is nondegenerate.

• The homogeneous quaternionic manifold $$(M,Q)$$ associated to an extended Poincaré algebra
• Bundles associated to the quaternionic manifold $$(M,Q)$$