Pseudo-immersions

Joris, Henri ; Preissmann, Emmanuel

Annales de l'Institut Fourier, Tome 37 (1987), p. 195-221 / Harvested from Numdam

Résumé

Si $f$ est un germe ${\cal C}^\infty $ de $({\bf R}^n,0)$, on dira que $f$ est une pseudo-immersion (on notera $f\in \Psi _{n,m}$) si tous les germes continus $g$ de $({\bf R},0)$ dans $({\bf R}^m,0)$, tels que $f\circ g\in {\cal C}^\infty $ sont eux-mêmes ${\cal C}^\infty $. On détermine complètement $\Psi _{n,1}$, et on montre que $\Psi _{2,2}={\rm Diff}_2$. Par ailleurs, si ${\bf K}={\bf R}$ ou ${\bf C}$ et si $g$ est une application de ${\bf K}$ dans ${\bf K}$ telle que $g^2$ et $g^3$ sont ${\cal C}^\infty $, alors $g$ est aussi ${\cal C}^\infty $. Si ${\bf K}={\bf H}$ (corps des hamiloniens) alors cette implication n’est plus vraie.

Abstract

Let $f:({\bf R}^m,0)\rightarrow ({\bf R}^n,0)$ be a ${\cal C}^\infty $-germ. $f$ is said to be a pseudo-immersion (noted $f\in \Psi _{n,m}$) if for continuous germ $g:({\bf R},0)\rightarrow ({\bf R}^m,0)$, $f\circ g\in {\cal C}^\infty $ implies $g\in {\cal C}^\infty $. $\Psi _{n,1}$, is completely determined, for each natural $n,\Psi _{2,2}$ is shown to coincide with ${\rm Diff}_2$. If ${\bf K}={\bf R}$ or ${\bf C}$ and $g:{\bf K}\rightarrow {\bf K}$ is such that $g^2$ and $g^3$ are in ${\cal C}^\infty $. If ${\bf K}={\bf H}$ (field of Hamiltonians), a counter-exemple shows that this implication is no more valid.

Détail

Publié le : 1987-01-01

MR 88e:57028 | Zbl 0596.58004

DOI : https://doi.org/10.5802/aif.1092

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