Remark on Goldschmidt’s Theorem on Formal Integrability

In this paper we apply Weil bundle techniques to the study of formal integrability for systems of nonlinear partial differential equations. We clarify the role of the curvature and show that both the statement and proof of Goldschmidt’s criterion on formal integrability are of algebraic nature. This fact allows us to obtain a version of this theorem which holds for smooth, algebraic, or analytic manifolds. Finally, we give a definition for the characteristic co-vectors of a system of partial differential equations and how their relationship with the Cauchy Kowalevski normal form.