Methods to Prove the h-Principle
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
KeywordsManifold Stratification Hull Stein Compressibility
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