非负截面曲率流形的几何与拓扑

Curvature and topology is one of central topics in Riemannian geometry. In this project we mainly study geometry and topology of manifolds with nonnegative sectional curvature and related problems. More precisely, we will make a comprehensive application of convergence theory of Riemannian manifolds, various topological tools and theory of isometric polar actions to attack the following problems:.1. Try to solve a problem proposed by Gromov and then construct a new class of special mapping from manifolds with nonnegative sectional curvature, which will be applied to study the topology of such manifolds and related important problems. .2. Construct a special class of mapping from some polar manifolds, which will be used to study the topology of such manifolds and related conjectures such as Grove-Ziller conjecture.

曲率与拓扑是黎曼几何的核心方向之一。在本项目中，我们主要研究非负截面曲率流形的几何与拓扑以及相关问题。具体来讲，我们将充分应用黎曼流形的收敛理论，拓扑学的诸多工具以及等距polar作用的理论深入研究以下问题：.1. 拟解决Gromov提出的一个问题，从而构造非负截面曲率流形上一类新的特殊映射，进一步利用它研究该类流形的拓扑以及相关重要问题；.2. 构造一类polar流形上的特殊映射，进一步利用它研究该类流形的拓扑以及相关重要猜想如Grove-Ziller猜想。

本项目主要研究了曲率与拓扑，群作用以及刚性问题。特别地，研究了曲率与Morse-Novikov 上同调群的关系，得到了几个新的消灭定理。从辛几何的角度研究了polar作用，并部分解决了辛几何中的Lerman-Montgomery-Sjamaar猜想。此外，给出了黎曼流形的曲率在某一点为常数的刻画，特别地得到了常曲率流形的一个新的刻画。迄今在Advances in Mathematics, Journal of Geometric Analysis, Archiv der Mathematik等杂志总共发表了三篇研究论文。