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Methods to Prove the h-Principle

  • Mikhael Gromov
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 9)

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: VX (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: VX (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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Mikhael Gromov
    • 1
  1. 1.Institute des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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