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A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures
Vicente Cortés, University of Bonn, Germany

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
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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.


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
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