Reflexion maps and geometry of surfaces in R^4
Journal of Singularities, Tome 21 (2020), pp. 84-96
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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.
@article{10_5427_jsing_2020_21e,
author = {Peter J. Giblin and Stanis{\l}aw Janeczko, and Maria Aparecida Ruas},
title = {Reflexion maps and geometry of surfaces in {R^4}},
journal = {Journal of Singularities},
pages = {84--96},
publisher = {mathdoc},
volume = {21},
year = {2020},
doi = {10.5427/jsing.2020.21e},
url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2020.21e/}
}
TY - JOUR AU - Peter J. Giblin AU - Stanisław Janeczko, AU - Maria Aparecida Ruas TI - Reflexion maps and geometry of surfaces in R^4 JO - Journal of Singularities PY - 2020 SP - 84 EP - 96 VL - 21 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.5427/jsing.2020.21e/ DO - 10.5427/jsing.2020.21e ID - 10_5427_jsing_2020_21e ER -
%0 Journal Article %A Peter J. Giblin %A Stanisław Janeczko, %A Maria Aparecida Ruas %T Reflexion maps and geometry of surfaces in R^4 %J Journal of Singularities %D 2020 %P 84-96 %V 21 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.5427/jsing.2020.21e/ %R 10.5427/jsing.2020.21e %F 10_5427_jsing_2020_21e
Peter J. Giblin; Stanisław Janeczko,; Maria Aparecida Ruas. Reflexion maps and geometry of surfaces in R^4. Journal of Singularities, Tome 21 (2020), pp. 84-96. doi: 10.5427/jsing.2020.21e
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