A new construction of homogeneous quaternionic manifolds and related geometric structures

1999/08/13 by Vicente Cortes, Cortes, Vicente
Mathematics · #53C25 #53C30 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C25 #msc:53C30

paper · pdf · doi:10.48550/arxiv.math/9908058

to appear in the Memoirs of the AMS, 81 pages, Latex source file

arxiv created 1999/08/13 · arxiv updated 2009/11/30

Abstract

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl⁰ (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge² W → V. If the skew symmetric vector valued bilinear form Πis 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. The construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any spin(V)-equivariant linear map Π: Sym² W → 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 Πis nondegenerate.

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