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Generic space curves and secants

Published online by Cambridge University Press:  14 November 2011

J. W. Bruce
J. W. Bruce
Affiliation:
School of Mathematics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU
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Synopsis

In this paper, we study the local structure of the secant mapping of a pair of disjoint curves. We show that for generic curves, the secant map and unit secant maps are locally stable. If we allow our curves to coincide, we can define anew unit secant map to be the natural unit tangent map near the diagonal. This is, for a generic curve, a locally stablemap away from the diagonal. Along the diagonal, it is locally stable as a ℤ2 symmetric germ (the ℤ2 symmetry originating with reflection in the diagonal).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 3-4 , 1984 , pp. 281 - 289
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Arnold, V. I.. Singularity Theory (London Math. Soc. Lecture Notes Series, No. 53) (Cambridge: Cambridge Univ. Press, 1981).CrossRefGoogle Scholar
2Bierstone, E.. Local properties of smooth maps equivariant with respect to finite group actions. J. Differential Geom. 10 (1975), 523–540.CrossRefGoogle Scholar
3Bruce, J. W.. On singularities, envelopes and elementary differential geometry. Math. Proc.Cambridge Philos. Soc. 89 (1981), 43–48.CrossRefGoogle Scholar
4Bruce, J. W. and Giblin, P. J.. Curves and Singularities (Cambridge: Cambridge Univ. Press, 1984).Google Scholar
5Bruce, J. W. and Giblin, P. J.. Smooth stable mappings on discriminant varieties. Proc. London Math. Soc., to appear.Google Scholar
6Golubitsky, M. and Guillemin, V.. Stable Mappings and Their Singularities (Graduate Texts in Maths) (Berlin: Springer, 1973).CrossRefGoogle Scholar
7Lipschutz, S.. Differential Geometry (Schaums Outline Series) (New York: McGraw-Hill, 1969).Google Scholar
8Martinet, J.. Singularities of Smooth Functions and Maps (London Math. Soc. Lecture Note Series, No. 58) (Cambridge: Cambridge Univ. Press, 1982).Google Scholar
9Mather, J.. Generic projections. Ann. of Math. 98 (1973), 226–245.CrossRefGoogle Scholar
10Poston, T. and Stewart, I. N.. Catastrophe Theory and its Applications (Surveys and Reference Works in Mathematics) (London: Pitman, 1978).Google Scholar