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 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 action 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.

Video de la plática
https://youtu.be/-0nK10guKOk

Foto de miniatura del video
https://photos.app.goo.gl/U9nXGbeubX7PXUBq1

Página del evento Singularities in generic geometry and its applications: Valencia V, para celebrar el 60 aniversario de Federico Sánchez Bringas
http://geogen.fciencias.unam.mx

Programa del evento
http://geogen.fciencias.unam.mx/On-line/program28JULIOIV.pdf

Resúmenes de las pláticas
http://geogen.fciencias.unam.mx/Abstracts/TalksandAbstracts.pdf

Cartel evento
https://goo.gl/photos/28EfgbZn6FyzVXEk7

Foto del evento
https://goo.gl/photos/2ZzpcGxaPW7M97n79

Paginas de Pedro Benedini
benedini@usp.br
https://www.dcce.ibilce.unesp.br/pos/webfacil/index.php?pg=84&Id_aluno=714

Artículos publicados de Pedro Benedini
https://arxiv.org/find/math/1/au:+Riul_P/0/1/0/all/0/1

Otras pláticas del evento Singularities in generic geometry and its applications: Valencia V, para celebrar el 60 aniversario de Federico Sánchez Bringas
https://www.youtube.com/playlist?list=PLiD-IJzweXR866QuMLRYm-FF-mCVogVRn

Página de de Ciencias TV
https://www.facebook.com/Ciencias-TV-170487220017476

Videos Académicos de (pláticas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqvXI_Bxb5iOH25gXprez6o

Videos de matemáticas en Ciencias TV
https://www.youtube.com/playlist?list=PLiD-IJzweXR8iyKKoKUGpAzF8rLWIu4A4

Videos de física publicados por Ciencias TV
https://www.youtube.com/playlist?list=PLiD-IJzweXR_vle91BMI7tnFH_uSkHuxo

Otros videos de Física y temas afines (platicas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqOEjKDh952uUoZe206qV7M

Otros videos de publicados por Ciencias TV
https://www.youtube.com/playlist?list=PLiD-IJzweXR_p5UtttIEN0sSTGfGeYpc-

Otros videos de publicados por Ciencias TV
https://www.youtube.com/playlist?list=PLiD-IJzweXR-ZEKDItWprRK9xXbv3i9tN

Otros videos de Física y temas afines (platicas grabadas)
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqOEjKDh952uUoZe206qV7M

Videos Académicos publicados
https://www.youtube.com/playlist?list=PLrg-5oUhFeirZBmxs7kDn_KrKHUJradjz

Videos Académicos publicados en 2017
https://www.youtube.com/playlist?list=PLrg-5oUhFeiqonNhVJlcrrcxHUrUyU2VA

Album de fotos miniatura de los videos Académicos publicados en 2017
https://goo.gl/photos/7yKCos9N8T9SecGk9

Agradecemos el apoyo de

Facultad de Ciencias de la UNAM
http://www.fciencias.unam.mx/
https://www.facebook.com/Facultad-de-Ciencias-214278861928417/?fref=ts

universo.math
http://universo.math.org.mx/

https://www.facebook.com/universo.math
Departamento de Matemáticas del CINVESTAV
http://www.math.cinvestav.mx/