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

具有下曲率下界的黎曼度量：在微分几何的基本问题中，确定存在的障碍或描述具有给定下曲率下界的流形的新例子。特别是，调查具有正的、非负的或几乎非负的截面曲率或Ricci曲率的度量的障碍或例子是有意义的。这里感兴趣的流形类是那些允许低上齐性的群作用的流形，特别是那些最多有两个上齐性的流形。辛连接和辛实现：连通和流形是对切向量沿可微路径的并行传输的描述。如果流形是辛的，那么如果并行传输保持辛形式，那么我们称连接为辛。我们研究了曲率满足一定条件的辛联络，即曲率的某一部分消失，或对其完整群的限制。有一种规范的方法可以在当地建立这种联系。这种方法是基于与四元数对称空间有关的某些Poisson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，从而确定何时可以在全局范围内推广局部障碍方法。特别是，重要的是要确定所涉及的泊松结构是否允许通过辛群群实现。

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.