Unstability problem of real analytic maps

TL;DR

This paper investigates the stability of real analytic maps, revealing that infinitesimal stability does not guarantee overall stability, with Whitney umbrellas exemplifying this discrepancy.

Contribution

It demonstrates that in the real analytic setting, infinitesimal stability is not sufficient for stability, and introduces a relative Whitney Approximation Theorem using Cartan's theorems.

Findings

Infinitesimal $C^ ext{omega}$ stability does not imply $C^ ext{omega}$ stability.

Whitney umbrellas are not $C^ ext{omega}$ stable.

A new relative Whitney Approximation Theorem is established.

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 $C^\omega$ stability does not imply $C^\omega$ stability; for instance, a Whitney umbrella $\mathbb{R}^2 \to \mathbb{R}^3$ is not $C^\omega$ 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.