# A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures

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*Vicente Cortés*

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.

#### Table of Contents

# Table of Contents

## A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures

- Contents vii8 free
- Abstract viii9 free
- Introduction 110 free
- 1. Extended Poincaré algebras 716 free
- 2. The homogeneous quaternionic manifold (M, Q) associated to an extended Poincaré algebra 1726
- 3. Bundles associated to the quaternionic manifold (M, Q) 3443
- 4. Homogeneous quaternionic supermanifolds associated to superextended Poincaré algebras 3948
- Appendix. Supergeometry 4453
- Bibliography 5968