Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2

Anton S. Galaev

Published 2006-12-14Version 1

The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-K\"ahlerian and special pseudo-K\"ahlerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives the classification of holonomy algebras for pseudo-K\"ahlerian manifolds of index 2. Finally we consider some examples and applications. We describe examples of 4-dimensional Lie groups with left-invariant pseudo-K\"ahlerian metrics and determine their holonomy algebras. We use our classification of holonomy algebras to give a new proof for the classification of simply connected pseudo-K\"ahlerian symmetric spaces of index 2 with weakly-irreducible not irreducible holonomy algebras. We consider time-like cones over Lorentzian Sasaki manifolds. These cones are also pseudo-K\"ahlerian manifolds of index 2. We describe the local DeRham-Wu decomposition of the cone in terms of the initial Lorentzian Sasaki manifold and we describe all possible weakly-irreducible not irreducible holonomy algebras of such cones.