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

具有曲率下界的黎曼度量：微分几何中的基本问题之一是确定具有给定曲率下界的流形的存在或描述新例子的障碍。特别是，研究具有正、非负或几乎非负截面或里奇曲率的度量的障碍或示例是很有趣的。这是感兴趣的流形类在这里是那些承认一组行动的低cohomogeneity，特别是那些cohomogeneity在最多两个。辛连接和辛实现：一个连接一个流形是描述的平行运输的切向量沿着可微路径。如果流形是辛的，那么我们称连接是辛的，如果平行迁移保持辛形式。我们研究辛连接的曲率满足一定的条件，即要么消失的某些部分的曲率，或限制其holonomy组。有一种规范的方法可以在本地构建这样的连接。该方法是基于与四元数对称空间相关的某些Poisson结构的辛实现的局部存在性。我们的目的是确定在何种情况下可以有一个全球实现，因此，当当地的障碍方法可以扩展到全球。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广群实现。

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.