On ricci curvature

This thesis is organized into six chapters. The first chapter provides a general introduction. The second chapter establishes the fundamental definitions and theorems in three-dimensional Euclidean and Lorentzian spaces. The third chapter is devoted to selected applications of Ricci curvature. In the fourth chapter, the theory of limit spaces is approached through the fundamental tools of Riemannian geometry. In this framework, geodesics, Riemannian distance, the Laplacian, and the Bochner formula are introduced, followed by a discussion of comparison techniques related to Ricci curvature. Fundamental results such as the Myers and Bishop–Gromov theorems are highlighted, and the geometric structures of manifolds with nonnegative Ricci curvature are examined. Furthermore, the structure of limit spaces of manifolds with Ricci curvature bounded from below is analyzed, emphasizing the decisive role of Ricci curvature in understanding both local and global geometric properties. The fifth chapter provides a detailed treatment of Ricci curvature within the context of comparison geometry. Finally, the sixth chapter is devoted to a discussion of the results and concluding remarks.