A GLOBAL THEORY OF SOME NONLINEAR CAUCHY-RIEMANN SYSTEM IN SEVERAL COMPLEX-VARIABLES

A system of nonlinear differential equations of the type partial deriv ative<(xj)over bar>w = <(f(j)(z,w))over bar>, j = 1,..., n, on a domai n of C-n is studied. Functional relations between the f(j)'s, j = 1,.. ., n, and other necessary conditions are deduced when at each point of the domain the system has a manifold of local solutions. A structure theorem, that makes possible to reduce the problems of the system, e.g . the global solvability of it, to the corresponding questions for a c onnection of the type partial derivative (z) over bar w = <(g(z, w))ov er bar> in a fibre bundle over a Riemann surface is proved, and throug h this reduction we obtain theorems of identity, extension, global fac torization, and so on, for the solutions of the system. As an example, a system of nonlinear differential equations of the type partial deri vative<(zj)over bar>w = <(a(j)(z))over bar>.<(p(w))over bar>, j = 1,.. ., n, is studied and its global solutions are constructed.