Holonomy groups of pseudo-quaternionic-K\"ahlerian manifolds of non-zero scalar curvature

Abstract

The holonomy group $G$ of a pseudo-quaternionic-K\"ahlerian manifold of signature $(4r,4s)$ with non-zero scalar curvature is contained in \Sp(1)\cdot\Sp(r,s) and it contains \Sp(1). It is proved that either $G$ is irreducible, or $s=r$ and $G$ preserves an isotropic subspace of dimension $4r$, in the last case, there are only two possibilities for the connected component of the identity of such $G$. This gives the classification of possible connected holonomy groups of pseudo-quaternionic-K\"ahlerian manifolds of non-zero scalar curvature.Comment: 7 pages; Dedicated to Dmitri Vladimirovich Alekseevsky at the occasion of his 70th birthda