Preface

Summary

The aim of this book is to introduce the reader to a geometric study of partial differential equations of second order.

We begin the book with the most classical subject: the geometry of ordinary differential equations, or more general, differential equations of finite type. The main item here is the various notions of symmetry and their use in solving a given differential equation. In Chapter 1 we discuss the distributions, integrability and symmetries. In a form appropriate to our aims, we remind the reader of the main notions of the geometry of distributions: complete integrability, curvature, integral manifolds and symmetries. The Frobenius integrability theorem is presented in its geometric form: as a flatness condition for the distribution.

The main result of this chapter is the famous Lie–Bianchi theorem which gives a condition and an constructive algorithm for integrability in quadratures of a distribution in terms of a Lie algebra of the shuffling symmetries. The theorem clearly explains the etymology of the expression “solvable Lie algebra.”

In Chapter 2 we apply these results to explicit integration of scalar ordinary differential equations.We consider some standard examples of differential equations integrable in quadratures but treat them in quite non-standard geometric way to demonstrate the advantage of the language and the method of symmetries. Even in the case of linear differential equations one is able to find some new and interesting results by systematically exploiting the notion of symmetries.