On exterior derivatives and solutions of ordinary differential equations

Abstract

Part I is concerned with sufficient conditions for the local uniqueness of solutions of nonlinear systems of ordinary differential equations and with necessary and sufficient conditions for the C' character of general solutions. Parts II and III give applications of these results. It is shown, in Part II, that the theorems of Part I imply those of [4; 8] on the local uniqueness of geodesics and on the introduction of geodesic coordinates for a binary positive definite Riemannian metric. In fact, Part I makes possible the extensions of these theorems from 2 to n dimensions and from geodesics to solutions of more general Euler-Lagrange systems. In Part III, the theorems of Part I are used to prove a theorem of Frobenius, dealing with systems of Pfaffians, under relaxed conditions of differentiability.