About the geometry of corank 1 surfaces in R4

Plática dada por Pedro Benedini Riul (ICMC/USP, Brazil) en el evento Singularities in generic geometry and its applications: Valencia V, para celebrar el 60 aniversario de Federico Sánchez Bringas el viernes 4 de agosto del 2017 en el Auditorio Carlos Graef del Amoxcalli en la Facultad de Ciencias de la UNAM

Abstract:
This talk is divided in two parts. In the first one, we present a more general study about the geometry of surfaces in R4 with corank 1 singularities. At the singular point we define the curvature parabola using the first and second fundamental forms of the surface. R.Mendes and J.J. NuñoBallesteros give in [5] a partition in four orbits of all corank 1 map germs f : (R2 , 0) → (R4 , 0) (see [2]) acording to their 2-jets under the of A2. We show that the curvature parabola distinguishes the four types of corank 1 singularities only by looking at the type of degeneracy of the parabola. We also show that two corank 1 2-jets (R2 , 0) → (R4 , 0) are equivalent under the of the subgroup R2 × O(4) iff there exists an isometry between the normal planes preserving the respective curvature parabola. Therefore, the curvature parabola contains all the local second order geometrical information of the surface. Inspired by [4] and [3] the definitions and some results
about asymptotic directions and umbilic curvature are given. The second part is dedicated to the classification of submersions (R4 , 0) → (R, 0) up to change of coordinates in the source that preserve the model surface X given 7 by (x, y) 7→ (x, xy, y2 , y3). Those change of coordinates form a geometric subgroup R(X) of the Mather R. This method is used in [1] and [6], where the authors classify submersions (R3 , 0) → (R, 0) that preserves the crosscap and the cuspidal edge, respectively. Using this classification we study the generic geometry related to the contact between X and hyperplanes in R4. Such contact is measured by the singularities of the height function of X. The results are part of my PhD thesis supervised by M. A. S. Ruas and it is a joint work with M. A. S. Ruas and R. Oset Sinha. Acknowledgements: We would like to thanks CAPES for financial support.

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https://arxiv.org/find/math/1/au:+Riul_P/0/1/0/all/0/1

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