Nonlinear mappings that are globally equivalent to a projection

The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse, F4 (x), to the derivative map F'(x) and a global solution to the differential equation P'(t) = Fl(P(t))(y — y0). Both these problems are solved and the generalized Hadamard-Levy criterion(formula present) is shown to be sufficient for F to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.