Manifolds with sectional and Ricci curvature bounds

本文研究了曲率界对具有不同曲率界的黎曼流形的局部几何结构和整体拓扑的影响以及它们的Gromov-Hausdorff极限的结构.（几乎）非负曲流形的结构及曲率下界的坍缩;2.具有Ricci曲率下界的流形的基本群; 3. Alexandrov空间和流形的亚度量;4.开的、负的和非正的收缩流形的结构。比较几何在理解黎曼流形的几何和拓扑之间的相互作用中起着重要的作用。最近的一个事实说明了这一点， 在庞加莱猜想的所谓解决方案G.~佩雷尔曼 我们提出了几个新的方法和研究方向，这可能会导致更深入的了解空间与各种曲率的界限。 这一点很重要，因为曲率，特别是里奇曲率，在数学和物理中起着核心作用。

This proposal is concerned with the effect ofcurvature bounds on the local geometric structure and global topology ofRiemannian manifolds with  various curvature bounds and thestructure of their Gromov-Hausdorff limits.The principal investigator will continue to study1. Structure of (almost) nonnegatively curved manifolds and collapsing with lower curvature bounds;2. Fundamental groups of manifolds with Ricci curvature bounded below;3.Submetries of Alexandrov spaces and manifolds;4. Structure of open negatively and nonpositively pinched manifolds.Comparison geometry plays an important role in understanding the  interplay between geometry and topology of Riemannian manifolds. This has recently been illustrated by the fact that it played an important role   in the purported resolution of Poincare conjecture by G.~Perelman.  We propose several new methods and directions of study which could lead to a greater insight about spaces with various curvature bounds.  This is important because curvature and Ricci curvature in particular plays a central role in mathematics and physics.