Partial Differential Relations pp 48-220 | Cite as

# 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

## Keywords

Vector Bundle Isometric Immersion Stein Manifold Differential Relation Manifold Versus## Preview

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