On Global Solutions for Partial Differential Equations of First Order

In this note we state a theorem which guarantees the existence and uniqueness of a global solution for the Cauchy initial value problem for a complete, regular system of partial differential equations of first order for one unknown function on a manifold—all data being of class C. This result depends on one hand on an investigation of the following problem for a w-dimensional C°°-manifold W, which is foliated by r-dimensional leaves, the leaf passing through wÇ: W being denoted by C(W)i Given a &-dim. submanifold A of W with k+r^m, find a foliated C°°-manifold S and a C°°-immersion j : S—>W such that (i) j(S) = \JaeA C(a), (ii) each leaf of S is mapped under j onto a leaf of W. Concerning this problem we prove: If for all a^A the tangent spaces of A and C(a) a t the point a have only the zero vector in common and if—in case r>l—for all aÇzA the leaf CN is a C°°-immersion. DATA AND DEFINITIONS. Let M be a, J-dim. manifold and T*(M) its cotangent-bundle. Consider the (2^ + 1)-dim. product-manifold T*(M)XR, where R denotes the real line. Let W be a system of r partial differential equations of first order f or one unknown f unction on M, i.e., W is a submanifold of T*(M)XR of codimension r. A local