New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures (eBook)

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-Kaehler manifold. If the metric $g$ is positive definite, i.e. a Riemannian metric, then the quaternionic Kaehler 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 Kaehler 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-Kaehler 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-Kaehler supermanifold $(M,Q,g)$ if the symmetric vector valued bilinear form $/Pi$ is nondegenerate.