Reflexion maps and geometry of surfaces in R4

Peter J. Giblin , Stanisław Janeczko , Maria Aparecida Ruas

Abstract

In this article we introduce new affinely invariant points—‘special parabolic points’— on the parabolic set of a generic surface M in real 4-space, associated with symmetries in the 2-parameter family of reflexions of M in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the special points, where the family has an additional degree of symmetry. Other points of M which are detected by the family of reflexions include inflexion points of real and imaginary type, and the first of these is also associated with sign changes on the parabolic set. We show how to compute the special points globally for the case where M is given in Monge form and give some examples illustrating the birth of special parabolic points in a 1-parameter family of surfaces. The tool we use from singularity theory is the contact classification of certain symmetric maps from the plane to the plane and we give the beginning of this classification, including versal unfoldings which we relate to the geometry of M.

Author Peter J. Giblin - [University of Liverpool]
Peter J. Giblin,,
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, Stanisław Janeczko (FMIS / DAST) - [Institute of Mathematics (IM PAN) [Polish Academy of Sciences (PAN)]]
Stanisław Janeczko,,
- Department of Analysis and Sigularity Theory
- Instytut Matematyczny Polskiej Akademii Nauk
, Maria Aparecida Ruas - [Universidade de Sao Paulo - USP]
Maria Aparecida Ruas,,
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Journal seriesJournal of Singularities, ISSN 1949-2006
Issue year2020
Vol21
Pages84-96
ASJC Classification2604 Applied Mathematics; 2608 Geometry and Topology
DOIDOI:10.5427/jsing.2020.21e
URL http://www.journalofsing.org/volume21/article5.html
Languageen angielski
Score (nominal)40
Score sourcejournalList
ScoreMinisterial score = 40.0, 12-08-2020, ArticleFromJournal
Publication indicators Scopus Citations = 0; Scopus SNIP (Source Normalised Impact per Paper): 2018 = 0.516
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