Unstability problem of real analytic maps

As well known, the C∞$C^\infty$ stability of proper C∞$C^\infty$ maps is characterized by the infinitesimal C∞$C^\infty$ stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal Cω$C^\omega$ stability does not imply Cω$C^\omega...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2024-10, Vol.56 (10), p.3174-3180
Main Authors: Bekka, Karim, Koike, Satoshi, Ohmoto, Toru, Shiota, Masahiro, Tanabe, Masato
Format: Article
Language:English
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Summary:As well known, the C∞$C^\infty$ stability of proper C∞$C^\infty$ maps is characterized by the infinitesimal C∞$C^\infty$ stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal Cω$C^\omega$ stability does not imply Cω$C^\omega$ stability; for instance, a Whitney umbrella R2→R3$\mathbb {R}^2 \rightarrow \mathbb {R}^3$ is not Cω$C^\omega$ stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms.13124