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<p>Smooth maps can represent manifolds in lower-dimensional spaces, particularly by their critical value sets, which are also called the discriminant sets. A twice folding map of the product of two Euclidean spaces to the plane, which is a smooth map defined by using a quadratic form on each space, is a useful piece to construct smooth maps for this purpose. It is, however, not a generic smooth map, and hence, the discriminant set of a map having it as a piece informs us less about the source manifold. In this article, we provide a special type of its perturbation into a generic smooth map <i>F</i> and study the shape of the discriminant set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(D_F\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation>, especially focusing on the relation to the images of the two Euclidean spaces, indices of the singular points, and types of the fibers. In Appendix, the classification of the fibers of <i>F</i> up to ambient diffeomorphism in the source dimension 4 is given.</p>

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A generic folding of the Euclidean n-space to the plane by two quadratic forms

  • 期刊论文

  • 发表日期：

    2026年1月6日

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  • Mahito Kobayashi,
  • Minoru Yamamoto

摘要

Smooth maps can represent manifolds in lower-dimensional spaces, particularly by their critical value sets, which are also called the discriminant sets. A twice folding map of the product of two Euclidean spaces to the plane, which is a smooth map defined by using a quadratic form on each space, is a useful piece to construct smooth maps for this purpose. It is, however, not a generic smooth map, and hence, the discriminant set of a map having it as a piece informs us less about the source manifold. In this article, we provide a special type of its perturbation into a generic smooth map F and study the shape of the discriminant set \(D_F\) D F , especially focusing on the relation to the images of the two Euclidean spaces, indices of the singular points, and types of the fibers. In Appendix, the classification of the fibers of F up to ambient diffeomorphism in the source dimension 4 is given.