This is a worth-while and successful attempt to introduce the ideas of Sophus Lie about foundations of PDE theory, which were not recognized as deserved for a century, into contemporary differential geometry. By the way, Lie’s methods and calculations are simplified by use of the framework of differential geometry developed in the second half of the last century. Lie’s main tool of classifying and solving differential equations was put into the background in the past by the overwhelming algebraic and functional analytic interest in Lie algebras, Lie groups, and its representations.
The author presents a pure local theory, as Lie did, restricted to local solutions, local solvability and local properties, but everything is discussed in the smooth, \(C^\infty\)-case, putting aside all profits of power series in the analytic case. There are no considerations whether and how local solutions may be glued together to global ones.
In the main part of the book, discussing PDE systems of second order in one dependent variable, the author restricts his considerations to the case of two or three independent variables. Here he shows how local Lie groups and local Lie pseudogroups dominate the local theory.
The book can be highly recommended to everyone, also to students, with interest in differential equations, provided the reader has a profound basic knowledge in differential geometry.
The author remarks at the end of his preface: “Most topics treated here can be found in the classical works of Lie, Cartan, and Vessiot. However a great effort has been made to present the ideas in a unified and very simple manner. In order not to obscure the fundamental issues it has been left to the reader to fill in the details needed to obtain his or her desired level of rigour.”
The book is organized in 18 chapters. 1. Introduction and summary. In Chapter 2, “PDE systems, Pfaffian systems and vectorfield systems”, the author starts with ODE systems and first-order PDE systems in one dependent variable, describing how the problem of solving a system is transformed into the question of finding integral manifolds of a Pfaffian system, or dually a vector field system defined on a submanifold of an appropriate jet bundle. The Frobenius theorem solves the problem in the simplest case of a complete vector bundle system. The general case is solved in Chapter 3 “Cartan’s local existence theorem”; Involutions and complete subsystems are introduced and maximal involutions and integrable vector fields are discussed. In Chapter 4, “Involutivity and the prolongation theorem”, the author sketches the prolongation theorem of Cartan and Janet and explains its consequences.
Chapter 5 “Drach’s classification, second-order PDE’s in one dependent variable, and Monge characteristics”, deals with a single PDE of second-order, singular vector fields and hyperbolic or parabolic PDE’s, and of course all things mentioned in its title.
Chapter 6, “Integration of vector field systems \(V\) satisfying \(\dim V'= \dim V+1\)”; \(V'\) denotes the derived vector field system. First-order PDE systems are discussed as a special case. The author shows that the theory of first-order PDE’s in one unknown is equivalent to the theory of contact transformations, hence contact transformations and Lie pseudogroups play an important role.
From this point on the following discussions on second-order PDE systems in this book are restricted to the cases of two or three independent variables.
Chapter 7. “Higher-order contact transformations”; is devoted to Lie’s rectification theorem for first-order PDE systems and to Bäcklunds two theorems on pseudogroups of contact transformations, referring the reader with respect to Bäcklund transformations and his theory for a special class of PDE systems to the literature.
Now it seems to be time to deal with local Lie groups as a special type of Lie pseudogroups. Chapter 8 “Local Lie groups” includes all fundamental notions and theorems as f.i. one parameter subgroups, structure constants, Maurer-Cartan forms, exponential mapping, and the three fundamental theorems of Lie.
Chapter 9, “Structural classification of 3-dimensional Lie algebras over the complex numbers”, gives the necessary background to study hyperbolic second-order PDE’s. In order to avoid analyticity arguments of Cartan’s local existence theorem, the author discusses the original context of Lie to integrate a complete vector field system by use of symmetry conditions as a preparation for the study of hyperbolic second-order PDE’s in Chapter 12. Chapter 10 “Lie equations and Lie vector field systems”. Chapter 11 “Second-order PDE’s in one dependent and two independent variables”, is devoted to the characterization of vector field systems arising from second-order PDE’s, and to a sketch of the Darboux method to find solutions by means of first integrals of the Monge systems.
As already mentioned Chapter 12, “Hyperbolic PDE’s with Monge systems admitting two or three first integrals”, gives an account of Vessiot’s theory of hyperbolic PDE’s in one dependent and two independent varaibles for which each of the Monge systems admits at least two independent first integrals, which includes the Goussat equations. It is shown that the finite number of canonical forms in the cassification obtained depends on the classification of 2- and 3-dimensional Lie groups achieved in Chapter 9.
Chapter 13, “Classification of hyperbolic Goursat equations”. The canonical forms of hyperbolic Goursat equations are characterized, followed by general solutions of the Goursat equations associated to the abelian 2-dimensional Lie group, associated to the affine group in one variable, and associated to the projective group in one variable. Chapter 14 “Cartan’s theory of Lie pseudogroups”, gives the basic facts of this theory as a preparation for the discussion of Cartan’s equivalence theorem in the next chapter. Chapter 15, “The equivalence problem”, is a survey of Cartan’s solution of the equivalence problem in order to use it, following Cartan’s “five variable” paper, to classify those parabolic PDE’s for which the double Monge system admits at least two functionally independent first integrals.
This finally is done in Chapter 16 “Parabolic PDE’s and associated PDE systems” and in Chapter 17 “The equivalence problem for general 3-dimensional Pfaffian systems in five variables”. The last Chapter 18, “Involutive second-order PDE systems in one dependent and three independent variables, solved by the method of Monge”, summarizes Cartan’s work on second-order PDE systems in one dependent and three independent variables. There are special points how to deal with one single PDE, two PDE’s, three PDE’s, and four and five PDE’s. It turns out that in the case of two PDE’s all systems can be solved by a reduction to ODE systems.
A last point: How to go further? gives an outlook on questions and problems to be discussed and solved in the future.