Memoirs of the American Mathematical Society 2013; 195 pp; softcover Volume: 225 ISBN10: 0821887998 ISBN13: 9780821887998 List Price: US$86 Member Price: US$68.80 Order Code: MEMO/225/1056
 The aim of this paper is to analyze 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. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically selfcontained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view. Table of Contents  Introduction
 The geometric setting
 Some geometric examples related to oscillation theory
 On the solutions of the ODE \((vz')'+Avz=0\)
 Below the critical curve
 Exceeding the critical curve
 Much above the critical curve
 Bibliography
