no. 2
Pseudo-immersions
Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 195-221.

Si f est un germe 𝒞 ∞ de (R n ,0), on dira que f est une pseudo-immersion (on notera f∈Ψ n,m ) si tous les germes continus g de (R,0) dans (R m ,0), tels que f∘g∈𝒞 ∞ sont eux-mêmes 𝒞 ∞ . On détermine complètement Ψ n,1 , et on montre que Ψ 2,2 = Diff 2 . Par ailleurs, si K=R ou C et si g est une application de K dans K telle que g 2 et g 3 sont 𝒞 ∞ , alors g est aussi 𝒞 ∞ . Si K=H (corps des hamiloniens) alors cette implication n’est plus vraie.

Let f:(R m ,0)→(R n ,0) be a 𝒞 ∞ -germ. f is said to be a pseudo-immersion (noted f∈Ψ n,m ) if for continuous germ g:(R,0)→(R m ,0), f∘g∈𝒞 ∞ implies g∈𝒞 ∞ . Ψ n,1 , is completely determined, for each natural n,Ψ 2,2 is shown to coincide with Diff 2 . If K=R or C and g:K→K is such that g 2 and g 3 are in 𝒞 ∞ . If K=H (field of Hamiltonians), a counter-exemple shows that this implication is no more valid.

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     author = {Joris, Henri and Preissmann, Emmanuel},
     title = {Pseudo-immersions},
     journal = {Annales de l'Institut Fourier},
     pages = {195--221},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {2},
     year = {1987},
     doi = {10.5802/aif.1092},
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     mrnumber = {88e:57028},
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Joris, Henri; Preissmann, Emmanuel. Pseudo-immersions. Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 195-221. doi : 10.5802/aif.1092. https://aif.centre-mersenne.org/articles/10.5802/aif.1092/

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