Lorentzian and conformal manifolds with special holonomy

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Manifolds with special geometries can be described by their holonomy representation. The irreducible holonomy representations of (simply-connected) Riemannian and pseudo-Riemannian manifolds are well known and geometric implications are intensively studied. In the pseudo-Riemannian case a new type of holonomy representations appears, the weakly irreducible but non-irreducible ones, which are - contrary to the irreducible case - not completely classified and geometrically less understood. In the same way as the holonomy of a metric, the holonomy of a conformal structure is defined (using the unique normal conformal Cartan connection). In the first period of the project the complete classification of Lorentzian holonomy groups was achieved (Th. Leistner). The aim in the second part of the project is to study further the geometric structure and to construct geometric models of Lorentzian manifolds with special holonomy. Furthermore, we want to study systematically the holonomy of conformal structures - in particular in the Lorentzian case - and their geometric implications in conformal geometry.