Non-density of stable mappings on non-compact manifolds

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  • Shunsuke Ichiki
    Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Tokyo, Japan

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Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is compact. The aim of this paper is to complement Mather's result. Namely, we show that the set of stable mappings in the set of all mappings is never dense if the source manifold is non-compact. Moreover, as a corollary of Mather's result and the main theorem of this paper, we give a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is not necessarily compact.

10 pages, to appear in Pure and Applied Mathematics Quarterly

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