An existence theorem for hyperbolic systems of partial differential equations is used to obtain a generalization of the Cartan-Kahler theorem to non-analytic differential systems. In order to apply this theorem certain restrictions must be placed on the differential system considered. A C(k)-differential system S in r independent variables satisfying these conditions is said to be C(k)hyperbolic in the x(r)-direction. After making this definition the following theorem is proved. Let S be C(k)hyperbolic in the x(r)-direction with k > or = 4(r+1) and genus g > or = r. Suppose I sub (r-1) is an (r-1)-dimensional C(k)-integral submanifold of S. Then in a neighborhood of each regular point p contained in I sub (r-1) of S there exists an r-dimensional C(k)-integral submanifold containing I sub (r-1). In case r = 2 the differential system need only be C(1) for the theorem to hold. The concept of a moving frame is introduced and used to set up four applications of the above theorem to problems in surface theory. In the section on applications it is proved that locally a 2-dimensional C(k)-Riemannian manifold has a C(k)-isometric imbedding in 3-dimensional euclidean space for k > or = 1. It is shown that the theorem cannot be used to obtain a local C(k)-conformal equivalence between two surfaces. (Author)
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