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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.

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
Full Text: DOI
[1] Dulac, H. 1904.Recherches sur les points singuliers des équations diff’rentielles, série 2 Vol. 9, 1–125. J. de L’Ecole Poly.
[2] X. Gong, Thesis, in preparation.
[3] Houzel C., Géométrie analytiqut locale 1 13 (1960)
[4] Melrose R. B., Equivalence of glancing hypersurfaces 37 pp 165– (1976) · Zbl 0354.53033
[5] Melrose R. B., Equivalence of glancing hypersurfaces II 255 pp 159– (1981) · Zbl 0472.53045
[6] Moser J. K., M, Webster
[7] Moser J. K., Normal forms for real surfaces in C2near complex tangents and hyperbolic surface transformation 150 pp 255– (1983)
[8] Oshima, T. 1978.On analytic equivalence of glancing hypersurfaces, Vol. 28, 51–57. Univ. Tokyo. · Zbl 0382.53026
[9] Siegel C. L., integrals of canonical systems 42 pp 806– (1941)
[10] Webster S. M., Holomorphic symplectic normalization of a real function 19 pp 69– (1992) · Zbl 0763.58010
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