本项目研究的典则度量是黎曼流形上的Einstein度量及其自然推广，包括梯度Ricci孤立子和拟Einstein度量等，它们可统一看作光滑度量测度空间上的“Einstein度量”。.Einstein度量是流形上最自然的度量，二维和三维情形，Einstein度量一定具有常截面曲率。而四维则复杂得多，对复流形，1990年田刚分类了四维正数量曲率Kaehler-Einstein流形，2012年LeBrun分类了四维正数量曲率Hermitian, Einstein流形。但是对实流形，即使假设正截面曲率，人们依然知之甚少。.本项目第一步将研究正截面曲率四维Einstein流形的刚性、正数量曲率四维Einstein流形与复结构的关系，第二步将借鉴第一步的研究方法去研究四维梯度Ricci孤立子和拟Einstein流形的刚性。

The canonical metrics investigated in this project are Einstein metrics on Riemannian manifolds and their natural genealizations, including gradient Ricci solitons and quasi-Einstein metrics, which can be considered as "Einstein metrics" on smooth metric measure spaces...Einstein metrics are most natural Riemannian metrics on manifolds. In dimensions two and three, they must have constant sectional curvature. While in dimension four, they are much more complicated, for the complex setting, in 1990 Tian classified Kaehler-Einstein four-manifolds with positive scalar curvature, and in 2012 LeBrun classified Hermitian, Einstein four-manifolds with positive scalar curvature. For the real setting, however much less is known, even assuming positive sectional curvature. ..The project is divided into two steps. Step one I will investigate the rigidity of Einstein four-manifolds with positive sectional curvature, and the relationship between Einstein metrics with positive scalar curvature and complex structure. Step two, I will borrow ideas in Step one to study the rigidity of four-dimensional gradient Ricci solitons and quasi-Einstein manifolds.