Geometry of Lorentzian manifolds with special holonomy

Manifolds with special geometries can be described by their holonomy representation. The irreducible holonomy representations of Riemannian and pseudo-Riemannian manifolds are known. Geometric properties of the corresponding special geometries are currently intensively studied. In the pseudo-Riemannian case a new type of holonomy representations appears, the indecomposable, but non-irreducible ones. Each special indecomposable Lorentzian holonomy representation is non-irreducible. Contrary to the irreducible case, the indecomposable non-irreducible holonomy representations are not classified. The knowledge about the corresponding special geometric structures is rar. The aim of the project is the description of the possible holonomy representations in the Lorentzian case and the study of geometric properties of the corresponding special Lorentzian geometries including the behaviour of geometric differential equations for such manifolds.