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Divergence for the normalization of real analytic glancing hypersurfaces. (English) Zbl 0804.53080
Glancing hypersurfaces $$[F: f(x\xi)= 0$$, $$G: g(x,\xi)= 0$$ at the symplectic space $$(\mathbb{R}^{2n},\sum^ n_{i=1} d\xi_ i\land dx_ i)$$, i.e., $$\{f,g\}(p)= 0$$, $$df\land dg(p)\neq 0$$, $$\{f,\{f,g\}\}(p)\neq 0$$, $$\{g,\{g,f\}\}(p)\neq 0$$, $$p\in F\cap G)]$$ were introduced by R. B. Melrose, who proved that they may be reduced to the following normal form $$\{x_ 1= 0\}$$, $$\{\xi_ n= \xi^ 2_ 1+ x_ 1\}$$. It was shown by T. Oshima [Sci. Pap. Coll. Gen. Ed., Univ. Tokyo 28, 51-57 (1978; Zbl 0382.53026)] that there exist pairs of analytic glancing hypersurfaces which cannot be reduced to the Melrose normal form through convergent symplectic mappings. It was done for dimension $$n\neq 2$$. The author gives a new proof of Oshima’s results for all dimensions.

##### MSC:
 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory
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##### References:
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