中文摘要：

该文讨论了某一类特殊流形的形状问题，即当某些紧的黎曼流形上存在一个非平凡的共形向量场且数量曲率为常数时，研究在什么情况下该流形等距于欧式空间中的球面．另外还研究当黎曼流形的数量瞳率是非常数时相应的若干刚性定理．

英文摘要：

In this paper, we discuss a question about what condition can enforce a compact Riemannian manifold carrying a nontrivial conformal vector field and with a constant scalar curvature to be isometric to an Euclidean sphere. We also study a Riemannian manifold with nonconstant scalar curvature and obtain some corresponding rigidity theorems.