ON SOME ASPECTS OF OSCILLATION THEORY AND GEOMETRY

This thesis aims to discuss some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. In this respect, we prove new results in both directions. For instance, we improve on classical oscillation and nonoscillation criteria for ODE's, and we find sharp spectral estimates for a number of geometric differential operator on Riemannian manifolds. We apply these results to achieve topological and geometric properties. In the first part of the thesis, we collect some material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.