A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures

Buch | Softcover

63 Seiten

2000
American Mathematical Society (Verlag)
978-0-8218-2111-4 (ISBN)

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

Introduction Extended Poincare algebras The homogeneous quaternionic manifold $(M,Q)$ associated to an extended Poincare algebra Bundles associated to the quaternionic manifold $(M,Q)$ Homogeneous quaternionic supermanifolds associated to superextended Poincare algebras Appendix. Supergeometry Bibliography.

Erscheint lt. Verlag	30.8.2000
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	0-8218-2111-3 / 0821821113
ISBN-13	978-0-8218-2111-4 / 9780821821114
Zustand	Neuware