Holonomy and the Stability Theorems

Abstract

In this chapter F denotes a foliation of codimension n and class C ^r, r ≥ 1, of a manifold M ^m. Our objective is to study the behavior of the leaves near a fixed compact leaf F. By the transverse uniformity of F it is sufficient to study the first returns of leaves to a small transverse section Σ of dimension n passing through a point p ∈ F. For each closed path γ in F passing through p, these returns can be expressed by a local C ^r diffeomorphism of Σ, f _γ, with f _γ (p) = p and where for x ∈ Σ sufficiently near p, f _γ(x) is the first return “over γ” of the leaf of F which passes through x.