This thesis consists of three parts. Each part solves a geometric problem in geometric analysis using differential equations. The first part gives a rigidity result to high dimensional positive Einstein manifolds, by controlling the upper bound of the integration of Weyl tensor. Part of the idea of the second part came from the new weighted Yamabe invariant from [4]. According to the definition, we can show a rigidity theorem to highdimensional compact shrinking Ricci solitons. The third part is an analytical result to 4-dimensional Ricci solitons. By the Weitzenbock for Ricci solitons introduced in [5], we proved an integral version of the Weitzenbock formula.