Lorentzian and conformal manifolds with special holonomy

具有特殊几何的流形可以用它们的完整表示来描述。（单连通）黎曼流形和伪黎曼流形的不可约完整性表示是众所周知的，几何含义被深入研究。在伪黎曼情形下，出现了一种新的完整性表示，弱不可约但非不可约的表示，与不可约情形相反，它们没有完全分类，在几何上也不太被理解。与度规的完整性相同，共形结构的完整性被定义（使用唯一的正规共形Cartan联络）。在该项目的第一阶段，实现了Lorentzian holonomy群的完整分类（Th。Leistner）。第二部分的目标是进一步研究具有特殊完整性的Lorentz流形的几何结构和构造其几何模型。此外，我们想系统地研究共形结构的完整性-特别是在洛伦兹的情况下-和它们的几何含义在共形几何。

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.