在一定的曲率条件下对流形进行分类一直是几何学研究的中心课题之一。众所周知，曲率流（尤其是 Ricci 流）是研究流形分类的重要而有效的工具。由于曲率流在演化过程中通常会产生奇点，流形的拓扑可能会发生改变。因此，运用曲率流理论去解决几何问题的一个关键就是对 Ricci soliton 结构的理解。在本项目中，我们重点研究收缩和稳定 Ricci soliton 的几何性质，并在此工作的基础上尝试完成 Ricci soliton 的分类。更进一步，我们也将研究特定条件下 Kahler-Ricci soliton 的分类。这些工作将会大大帮助我们去了解 Ricci 流下流形的奇点结构，从而为完成某些曲率条件下流形的分类提供可能。

Classification of manifolds under certain restriction on the curvature is a central theme in the study of geometry. As we know, curvature flows (especially Ricci flow) are important and efficient tools to investigate the classification of manifolds. Since singularities will arise in the envolution of the curvature flows, the topology of manifolds would be changed. Thus, the key step to understand the geometry of manifolds using Ricci flow is to investigate the structure of Ricci soliton. In this project, we mainly investigate the geometry of shrinking and steady Ricci soliton and then try to classify the Ricci solitons. And furthermore, we will investigate the classification of Kahler Ricci soliton under suitable curvature conditions. This project will help us understand the structure of singularities under the Ricci flow and classify certain manifolds.