Stability of C ∞ Mappings: I. The Division Theorem

Abstract

Let f: Ni Pi (i = 1, 2) be Co mappings of Co manifolds. We will say fi and f2 are isomorphic if there exist Co diffeomorphisms h: N1 N2 and h': P1 P2 such that h' o f1 = fo h. The main notion studied in this series of papers will be that of (oo -structurally) stable mapping. Roughly speaking, a Co mapping f: N P is said to be (ao -structurally) stable if every Co mapping g: N o P which is sufficiently close to f in a suitable topology (the direct limit of the fine Ck topologies, as k Ao) is isomorphic to f (cf. Thom and Levine [7]). There is an infinitesimal notion corresponding to stability. Namely, we say a Co mapping f: N o P is infinitesimally stable if for every Co vector field w along f there exist Co vector fields u on N and v on P such that w = Tf a u + v o f. (Our terminology and notation is the following: TN denotes the tangent bundle of N. N, denotes the tangent space to N at n, for each n e N. A vector field along f is a mapping w: N TP such that w(n) e Pf(,) for all n e N. The mapping of tangent bundles induced by f is denoted Tf: TN-+ TP.) In Stability of Mappings: II, we will show that a proper, infinitesimally stable, Co mapping is stable. In a later paper we will show that a proper, stable, Co mapping is infinitesimally stable. Thus, a proper, Co mapping is stable if and only if it is infinitesimally stable. A very simple example will be given in Stability of Mappings: II to show that infinitesimal stability of f does not imply stability of f if the hypothesis that f is proper is dropped. Also, if this hypothesis is dropped, stability of f does not imply infinitesimal stability. B. Morin has constructed an example which shows this. The purpose of this paper is to state and prove an analytic result which will be used in our proof that a proper, infinitesimally stable, Co mapping is stable. The result is that there exist mappings Q and H satisfying a certain identity (equation (1) of ? 2) and having certain differentiability properties.