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

TL;DR

This paper classifies the possible connected holonomy groups of pseudo-quaternionic-K"ahlerian manifolds with non-zero scalar curvature, revealing conditions under which these groups are irreducible or preserve isotropic subspaces.

Contribution

It provides a complete classification of the connected holonomy groups for these manifolds, identifying when they are irreducible or preserve isotropic subspaces.

Findings

Holonomy group is contained in (1)(r,s)

Holonomy group is either irreducible or preserves an isotropic subspace when s=r

Two possible connected holonomy groups when preserving an isotropic subspace

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.