Groupoids and differentiable structures

This thesis consists of four chapters. In the first chapter, weestablish the problems studied in this thesis and give themotivation for studying these topics. We also recall main results obtained so far and outline our results.In the second chapter, we recall basic definitions,theorems and examples which are used in the rest of the thesis.Chapter 3 and chapter 4 are original parts of the thesis. In the third chapter, as a generalization of the notions of principal topological groupoids and principal $G$-spaces, we introduce the notion of principal Lie groupoids. We show that the category of principal $G$-manifolds and the category of principal Lie groupoids are equivalent. The results obtained in this thesis include the case of locally trivial Lie groupoids and compact Hausdorff Lie groupoids.In the fourth chapter, we consider the generalized paracomplex manifolds and Lie groupoids.We first give necessary and sufficient conditions for almost generalized paracomplex manifolds in terms of a bivector field and certain 2-forms. Then we investigate relationship between the conditions of a generalized paracomplex manifolds and Lie groupoids. We show that the integrability condition of generalized almost paracomplex manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic forms and paraholomorphic maps between generalized paracomplex manifolds are related to each other.KEY WORDS: Groupoid, G-manifold, principal G-manifold, Lie groupoid,principal Lie groupoid, symplectic groupoid, generalized almost paracomplex manifold.