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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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