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A deformation lemma on a C$$^ 1$$ manifold. (Un lemme de déformation sur une sous-variété de classes C$$^ 1$$.) (French. Abridged English version) Zbl 0801.57022
Let $$S = \{x \in E \mid g_ 1(x) = 1,\dots,g_ k(x) = 1\}$$ where $$g_ i \in C^ 1(E,R)$$, 1 is a regular value of $$g_ i$$ and where the family of $$k$$ submanifolds $$H_ i = \{g_ i = 1\}$$ $$i = 1,\dots,k$$ is transverse. Using also the numbers $$\| f'/s_ w \| = \inf_{\lambda_ 1,\dots,\lambda_ k\in R}\| f'(x) - \sum^ k_{i = 1} \lambda_ i g_ i' (x)\|$$ and $$t_ g (R) = \inf_{\lambda_ 1,\dots,\lambda_ R \in S^{k - 1}} \| \sum^ k_{i = 1} \lambda_ i g_ i'(x)\|$$ the author defines a condition similar to the Palais-Smale condition.
Then he proves a theorem which is some kind of deformation lemma, namely:
Let $$S = \{x \in E \mid g_ 1(x) = 1,\dots,g_ k(x) = 1\}$$ be a $$C^ 1$$ submanifold of a Banach space $$E$$, let $$f\in C^ 1(U,R)$$, where $$U$$ is a neighborhood of $$S$$. If $$f|_ S$$ and $$g_ i' s$$ satisfy the condition above at $$a$$ which is a regular value of $$|_ S$$, then there exists $$\widehat{\varepsilon} > 0$$ such that for all $$0 < \varepsilon < \widehat{\varepsilon}$$ there exists a homeomorphism $$h$$ of $$S$$ onto $$S$$ such that
a) $$h(x) = x$$ if $$f(x) \not\in [a - \widehat{\varepsilon}, a + \widehat{\varepsilon}]$$,
b) $$f(h(x))\leq f(x)$$ for all $$x \in S$$,
c) $$f(h(x))\leq a - \varepsilon$$ for all $$x$$ such that $$f(x) \leq a + \varepsilon$$,
d) If $$S$$ is symmetrical $$(S = -S)$$ and if $$f$$ is even then $$h$$ is odd.
The case where $$\dim E < \infty$$ is treated in a different way, and it is stronger. The author gives one outline of the proof for the case $$k = 1$$ and $$\dim E = \infty$$. He uses the integral lines of two pseudo-gradient vector fields for $$f$$ and $$g$$ in order to build the deformation $$h$$. The complete details of the results of the paper appear in Manuscr. Math. 81, No. 3-4, 339-359 (1993)].

##### MSC:
 57R70 Critical points and critical submanifolds in differential topology