典型黎曼流形的刚性与分类问题研究历来是微分几何的重要关切。近年来，随着几何分析方法、泛函临界点研究技巧在此类课题研究上的成功运用，大量典型 Ricci-Hessian 型流形不断涌现，其中既包括与曲率流密切相关的 Ricci soliton、广义 m-quasi-Einstein 流形等拟爱因斯坦流形，也包括 Miao-Tam 度量、CPE 度量等重要黎曼泛函在限制条件下的临界点。流形上 Ricci-Hessian 型方程往往在很大程度上刻画流形的本质，如 Obata 关于球面度量的经典刻画。在前期拟爱因斯坦流形的研究基础上，本项目将集中研究 Ricci-Hessian 型流形在一定前提条件下的分类及刚性问题，特别专注于具有特殊几何与拓扑性质的广义 m-quasi-Einstein 流形、h-almost Ricci soliton 、Miao-Tam 度量、CPE 度量的刚性与分类研究。

The investigations on rigidity and classification of typical Riemannian manifolds are important concerns of differential geometric theory since its beginning. In recent years, along with the successful applications of methods in those subjects like geometric analysis, theory of critical points on geometric functional, a number of concepts of Ricci-Hessian type manifolds are constantly emerging. These Ricci-Hessian type manifolds include not only quasi Einstein manifolds like Ricci soliton closely related to curvature flow, but some special critical points of the important Riemannian functionals under the restrictive conditions like Miao-Tam metric and CPE metric. The Ricci-Hessian type tensor equations can always characterize manifold essences to a great extent, for instance, the characterization of the standard sphere metric by Obata. In this project, relying on the previous research foundation on quasi Einstein manifolds, we will concentrate on studying problems related to rigidity and classification of Ricci-Hessian type Riemannian manifolds under some assumptions, especially focus on rigidity and classification of generalized m-quasi-Einstein manifold, h-almost Ricci soliton, Miao-Tam critical metric, CPE metric with typical geometric or topological properties.