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
KeywordsVector Bundle Isometric Immersion Stein Manifold Differential Relation Manifold Versus
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