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On Some Aspects of Oscillation Theory and Geometry
Bruno Bianchini, Universita Degli Studi Di Padova, Italy, Luciano Mari, Universidade Federal Do Ceara, Fortaleza, Brazil, and Marco Rigoli, University degli Studi di Milano, Italy
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Memoirs of the American Mathematical Society
2013; 195 pp; softcover
Volume: 225
ISBN-10: 0-8218-8799-8
ISBN-13: 978-0-8218-8799-8
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 self-contained, 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.

• 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