Geometric deformations of curves in the Minkowski plane

Author(s)

Alex Paulo Francisco

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2019-04-16

Examining board members

Farid Tari; Maria Elenice Rodrigues Hernandes; Miriam Garcia Manoel; Luciana de Fátima Martins

• Advisor: Farid Tari

Abstract

In this work, we extend the method developed in (SALARINOGHABI, 2016),(SALARINOGHABI; TARI, 2017) to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in detail with all local phenomena that occur generically in 2-parameters families of curves. In each case, we obtain the geometry of the deformed curve, that is, information about inflections, vertices and lightlike points. We also obtain the behavior of the evolute/caustic of a curve at special points and the bifurcations that can occur when the curve is deformed. Moreover, in order to obtain the generic deformations at a lightlike inflection point of order 2, we also classify submersions from R3 to R by diffeomorphisms in the source that preserve the swallowtail and, using such classification, we study the flat geometry of the swallowtail, which comes from its contact with planes, which in turn is measured by the singularities of the height function on the swallowtail. (AU)

• FAPESP's process: 15/16177-2 - Geometric deformations of curves in the Minkowski plane
• Grantee: Alex Paulo Francisco
• Support Opportunities: Scholarships in Brazil - Doctorate