Singularities of secant maps of immersed surfaces |
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Authors: | Sunayana Ghosh Joachim H Rieger |
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Institution: | 1. Institut für Algebra und Geometrie, Martin-Luther-Universit?t Halle, D-06099, Halle (Saale), Germany
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Abstract: | The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion $X\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c}The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under
the immersion. The germ of the secant map of a generic codimension-c immersion
X:\mathbb Rn ? \mathbb Rn+cX\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c} at the diagonal in the source is a
\mathbb Z2{\mathbb Z}_2 stable map-germ
\mathbb R2n ? \mathbb Rn+c-1{\mathbb R}^{2n} \to {\mathbb R}^{n+c-1} in the following cases: (i) c≥ 2 and (2n,n + c − 1) is a pair of dimensions for which the
\mathbb Z2{\mathbb Z}_2 stable germs of rank at least n are dense, and (ii) for generically immersed surfaces (i.e., n = 2 and any c≥ 1). In the latter surface case the
A\mathbb Z2{\mathcal A}^{{\mathbb Z}_2}-classification of germs of secant maps at the diagonal is described and it is related to the
A{\mathcal A}-classification of certain singular projections of the surfaces. |
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Keywords: | |
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