eBook ISBN:  9781470402914 
Product Code:  MEMO/147/700.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 
eBook ISBN:  9781470402914 
Product Code:  MEMO/147/700.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 147; 2000; 63 ppMSC: Primary 53
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.
ReadershipGraduate students and research mathematicians interested in differential geometry.

Table of Contents

Chapters

Introduction

1. Extended Poincaré algebras

2. The homogeneous quaternionic manifold ($M, Q$) associated to an extended Poincaré algebra

3. Bundles associated to the quaternionic manifold ($M, Q$)

4. Homogeneous quaternionic supermanifolds associated to superextended Poincaré algebras


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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.
Graduate students and research mathematicians interested in differential geometry.

Chapters

Introduction

1. Extended Poincaré algebras

2. The homogeneous quaternionic manifold ($M, Q$) associated to an extended Poincaré algebra

3. Bundles associated to the quaternionic manifold ($M, Q$)

4. Homogeneous quaternionic supermanifolds associated to superextended Poincaré algebras