Methods to Prove the h-Principle

Abstract

Consider a differential relation ℛ ⊂ X ^(r) whose complement Σ = X ^(r)\ℛ is a closed stratified subset in X ^(r) of codimension m ≥ 1 and take a generic holonomic C^∞-section f: V → X ^(r) whose singularity _{Σf = f} ⁻¹(Σ) ⊂ V may be non-empty (compare 1.3). Let us try to solve ℛ by deforming f to a holonomic Σ-non-singular section f: V → X ^(r). Such a deformation can not be, in general, localized near _Σf [see Exercise (a) below] but one can find in some cases an auxiliary subset Σ′ = Σ′(f) ⊃ _Σf in V of codimension m — 1, such that the desired deformation does exist in an arbitrarily small neighbourhood of Σ′. The major difficulty in the construction of f comes from the holonomy condition. In fact, the problem becomes quite easy without this condition, as one can see in the following