Unstability problem of real analytic maps

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

7 pages