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Unstability problem of real analytic maps

Karim Bekka, Satoshi Koike, Toru Ohmoto, Masahiro Shiota, Masato Tanabe

Abstract

As well-known, the $C^\infty$ stability of proper $C^\infty$ maps is characterized by the infinitesimal $C^\infty$ stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the infinitesimal $C^ω$ stability does not imply $C^ω$ stability; for instance, a Whitney umbrella $\mathbb{R}^2 \to \mathbb{R}^3$ is not $C^ω$ stable. A main tool for the proof is a relative version of Whitney's Analytic Approximation Theorem which is shown by using H. Cartan's Theorems A and B.

Unstability problem of real analytic maps

Abstract

As well-known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrella is not stable. A main tool for the proof is a relative version of Whitney's Analytic Approximation Theorem which is shown by using H. Cartan's Theorems A and B.

Paper Structure

This paper contains 8 sections, 7 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^\omega$ Stability
  4. Key facts in the $C^\omega$ category
  5. Proof of Theorem \ref{['main']}
  6. Relative version of Whitney's approximation theorem
  7. Necessary condition for $C^\omega$ stability
  8. Remark on a sufficient condition for $C^\omega$ stability

Key Result

Theorem 1.1

Let $M$ and $N$ be $C^\omega$ manifolds and $f \colon M \to N$ a proper $C^\omega$ map which is $C^\infty$ stable. If $f$ is not an immersion in case of $m < n$, or $f$ has an $A_3$-singular point (swallowtail singularity) in case of $m \geqslant n$, then $f$ is $C^\omega$ unstable.

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1
  • Corollary 2.2
  • Theorem 2.3: Whitney's Approximation Theorem Whitney
  • Theorem 2.4: Cartan's Theorem Cartan, see also FS
  • Theorem 2.5: Tognoli's Approximation Theorem Tognoli (see also BKS)
  • Lemma 3.1
  • Proof
  • Lemma 3.2
  • Proof