Lorentzian and conformal manifolds with special holonomy

具有特殊完整性的洛伦兹流形和共形流形

• 批准号: 5453236
• 负责人: Privatdozent Dr. Felipe Leitner
• 金额: --
• 依托单位: Department of Pure Mathematics
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2006-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5453236?language=en
• 关键词: Lorentzian; conformal; manifolds; special; holonomy

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.

具有特殊几何形状的流形可以通过其完整表示来描述。 （单连通）黎曼流形和伪黎曼流形的不可约完整表示是众所周知的，并且对其几何含义进行了深入研究。在伪黎曼情况下，出现了一种新类型的完整表示，即弱不可约但不可约的表示，与不可约情况相反，它们没有完全分类并且在几何上较少被理解。与度量的完整性相同，定义了共形结构的完整性（使用唯一的法向共形嘉当连接）。在该项目的第一阶段，实现了洛伦兹完整群的完整分类（Th. Leistner）。该项目第二部分的目的是进一步研究洛伦兹流形的几何结构并构建具有特殊完整性的洛伦兹流形的几何模型。此外，我们希望系统地研究共形结构的完整性——特别是在洛伦兹情况下——及其在共形几何中的几何含义。