Compactness theorems on Riemannian manifolds

In this thesis, first, basic definitions and theorems are introduced about Riemannian manifolds considered as a special metric space. With the aid of second variational of energy functional, Myers-type compactness theorems are proved. Next, linear second order oscillatory differential equations and Riccati comparison theorem are given. On using Riccati comparison theorem, new compactness theorems are acquired. Throughout the thesis all of the obtained compactness theorems contain specific assumptions concerning Ricci curvature tensor.