Exterior Differential Systems and its Applications

The first part of this book contains the theory of integration of total differential equations connected with a general system of exterior differential forms (covariant alternating quantities). The symbolism used is the w-method introduced in Cartans well-known publications [Ann. Sci. Ecole Norm. Sup. (3) 18, 24–311 (1901); 21, 153–206 (1904), in particular, chap. I] with some modifications due to E.Kahler [Einfuhrung in die Theorie der Systeme von Differential-gleichungen, Hamburger Math. Einzelschr., no. 16, Teubner, Leipzig-Berlin, 1934]. The first two chapters contain an exposition of the method. In chapter III, after introducing the important notions of closed systems and characteristic systems the theory of completely integrable systems is presented and applied to the ordinary problem of Pfaff. Chapter IV contains the definitions of the integral elements, the characters and the genus and two fundamental existence theorems. Systems in involution are defined in chapter V and this chapter contains several simple forms of the conditions for these systems. The theory of prolongation is dealt with in chapter VI. For the chief theorem of prolongation, proved by Cartan in 1904, another proof is given for the case m = 2. In No. 117 special attention is paid to the cases where the proof is not valid. The second part of the book contains applications to several problems of differential geometry. Chapter VII deals with old and new problems of the classical theory of surfaces. In each case the degree of freedom of the solution is discussed. The last chapter contains problems with more than two independent variables. A new method is developed for orthogonal systems in n variables. The problem of the realization of a V2 with a given ds2 in an R6 is discussed elaborately and special attention is paid to the singular solutions of this problem. Reviewed by J. A. Schouten