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Bifurcation sets of families of reflections on surfaces in ℝ3

Part of: Classical differential geometry Differential topology

Published online by Cambridge University Press:  14 February 2017

P. J. Giblin  and
S. Janeczko
P. J. Giblin
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK ([email protected])
S. Janeczko
Affiliation:
Instytut Matematyczny PAN, ul. Sniadeckich 8, 00-950 Warszawa, Poland and Wydzial Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Pl. Politechniki 1, 00-661 Warszawa, Poland ([email protected])
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Extract

We introduce a new affinely invariant structure on smooth surfaces in ℝ3 by defining a family of reflections in all points of the surface. We show that the bifurcation set of this family has a special structure at ‘ points’, which are not detected by the flat geometry of the surface. These points (without an associated structure on the surface) have also arisen in the study of the centre symmetry set; using our technique we are able to explain how the points are created and annihilated in a generic family of surfaces. We also present the bifurcation set in a global setting.

Keywords

affine invariantssurfacesparabolic setsbifurcation sets

MSC classification

Secondary: 53A05: Surfaces in Euclidean space 57R45: Singularities of differentiable mappings
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 2 , April 2017 , pp. 337 - 352
Copyright
Copyright © Royal Society of Edinburgh 2017 

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