C^1 Approximation of Vector Fields on the Renormalization Group Method and its Applications (繰り込み群によるベクトル場の C^1 近似とその応用) The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining solutions which are approximate to exact solutions for a long time interval. In this work, it is shown that for a given vector field on a manifold and for the differential equation associated with it, a family of approximate solutions obtained on the RG method defines a vector field which is close to the original vector field in C^1 topology under appropriate assumptions. Furthermore, some topological properties of the original vector field such as the existence of an invariant manifold and its stability are inherited from those for the RG equation. This fact is applied to show the presence of the Hopf bifurcation and other global bifurcation. The present RG method is also useful for detecting an approximate center manifold and a flow thereon. This method is viewed as strict mathematical formulation of the reductive perturbation method proposed by Kuramoto, and includes the geometric singular perturbation method proposed by Fenichel as a particular case. The RG method applied to time dependent linear equations shows that the stability of the trivial solutions are inherited from those of the RG equations. This fact results in the presence of synchronization in coupled systems.