数量曲率在微分几何及广义相对论的研究中一直居于重要的地位。自上世纪70年代开始的一系列对数量曲率形变的研究工作，揭示了其与著名的正质量定理之间深刻的联系。另一方面，作为另一个数量型曲率量代表的Q-曲率，对其黎曼几何性质的研究却并不多见。本项目试图从有关数量曲率的经典研究入手，深入探讨数量型曲率量的形变问题，论证数量曲率、Q-曲率以及一般的黎曼不变量具有深刻的内在相似性，从而发现并理解有关数量型曲率量在黎曼度量模空间上存在的丰富几何结构。

The study of scalar curvature plays an important role in both differential geometry and general relativity. A series of researches originated from 1970's on scalar curvature revealed deep connections between deformations of scalar curvature and positive mass theorem. On the other hand, as another typical scalar-type curvature, Q-curvature, there are not too many researches focusing on its Riemannian geometric properties. Starting from classical researches on scalar curvature, this project try to investigate problems on deformations of scalar-type curvature and show that scalar curvature, Q-curvature and generic Riemannian invariants share essentially similar geometric properties. With the aid of these results, we can reveal and understand the interesting geometric structure of the module space of all Riemannian metrics with respect to these scalar-type curvature.