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Some remarks on the equation \(f(x)= 0\) in \(\mathbb{R}{}^ n\). (English) Zbl 0757.58007

let \(f: D\to \mathbb{R}^ n\) be a smooth map on an open subset of \(\mathbb{R}^ n\). The author proves that, if \(df_ a\in\text{Isom}(R^ n)\) for all \(a\in D\), then \(f^{-1}(0)\) coincides with the set of critical points of a Morse function \(g: D\to R\) of a prescribed Morse index. The result is applied to describe analogously the set of fixed points of \(f\) under the hypothesis that \(df_ a-1_{R^ n}\in\text{Isom}(R^ n)\) for all \(a\in D\).
Reviewer: D.Motreanu (Iaşi)

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57R70 Critical points and critical submanifolds in differential topology
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