Riemannian metrics with lower curvature bounds. Special symplectic connections and symplectic realizations.

具有下曲率界限的黎曼度量。

• 批准号: 5406882
• 负责人: Professor Dr. Lorenz Schwachhöfer
• 金额: --
• 依托单位: Lehrstuhl VII: Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5406882?language=en
• 关键词: Riemannian; metrics; lower; curvature; bounds.

Riemannian metrics with lower curvature bounds: Among the fundamental questions in differential geometry is the determination of obstructions to the existence or the description of new examples of manifolds with given lower curvature bounds. In particular, it is of interest to investigate obstructions to or examples of metrics with positive, nonnegative or almost nonnegative sectional or Ricci curvature. The class of manifolds which are of interest here are those which admit a group action of low cohomogeneity, in particular those of cohomogeneity at most two.Symplectic connections and symplectic realizations: A connection an a manifold is the description of the parallel transport of tangent vectors along differentiable paths. If the manifold is symplectic, then we call the connection symplectic if parallel transport preserves the symplectic form. We investigate symplectic connections whose curvature satisfies certain conditions, namely either the vanishing of some part of the curvature, or restrictions on its holonomy group. There is a canonical method to construct such connections locally. This method is based on the local existence of symplectic realization of certain Poisson structures related to quaternionic symmetric spaces. The aim is to determine in which case there can be a global realization and hence when the local obstruction methods can be extended globally. In particular, it is important to decide if the Poisson structures involved admit a realization by a symplectic groupoid.

具有下曲率界的黎曼度量：微分几何中的基本问题之一是确定存在的障碍或描述具有给定下曲率界的流形的新例子。特别地，研究具有正、非负或几乎非负截面曲率或里奇曲率的度量的障碍或例子是有趣的。我们感兴趣的一类流形是那些具有低同质性的群作用的流形，特别是那些最多只有两个同质性的流形。辛连接和辛实现：流形上的连接是对切向量沿可微路径平行移动的描述。如果流形是辛的，那么我们称这种连接是辛的，如果平行移动保持辛形式。我们研究了曲率满足某些条件的辛连接，即曲率的某些部分消失或对其完整群的限制。有一种规范的方法可以在局部构造这样的连接。该方法基于与四元数对称空间相关的泊松结构的辛实现的局部存在性。目的是确定在哪种情况下可以实现全局实现，从而确定何时可以对局部阻塞方法进行全局扩展。特别地，重要的是决定所涉及的泊松结构是否允许一个辛类群来实现。