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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2021.tde-27082021-141339
Document
Author
Full name
Tito Alexandro Medina Tejeda
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2021
Supervisor
Committee
Ruas, Maria Aparecida Soares (President)
Figueiredo Junior, Ruy Tojeiro de
Ishikawa, Goo
Martins, Luciana de Fátima
Title in English
A new approach to the differential geometry of frontals in the Euclidean space
Keywords in English
Front
Frontal
Gaussian curvature
Mean curvature
Principal curvatures
Abstract in English
In this work we investigate the differential geometry of singular surfaces known as frontals. We prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the frontals in Euclidean 3- space. Also, we characterize in a simple way these singular surfaces and its fundamental forms with local properties in the differential of its parametrization and decompositions in the matrices associated to the fundamental forms. In particular we introduce new types of curvatures which can be used to characterize wave fronts. On the other hand, we investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities of fronts. Furthermore, we study the convergence to infinite limits of these geometrical invariants and we show how this is tightly related to a property of approximation of fronts by parallel surfaces.
Title in Portuguese
Uma nova abordagem da geometria diferencial de frontais no espaço euclidiano
Keywords in Portuguese
Curvatura Gaussiana
Curvatura Média
Curvaturas Principais
Frente
Frontal
Abstract in Portuguese
Neste trabalho investigamos a geometria diferencial de superfícies singulares conhecidas como frontais. Provamos um resultado semelhante ao teorema fundamental das superfícies regulares na geometria diferencial clássica, que estende o teorema clássico aos frontais no espaço Euclidiano. Além disso, caracterizamos de forma simples essas superfícies singulares e suas formas fundamentais com propriedades locais na diferencial de sua parametrização e decomposições nas matrizes associadas às formas fundamentais. Em particular, introduzimos novos tipos de curvaturas que podem ser usadas para caracterizar as frentes de onda. Por outro lado, investigamos as condições necessárias e suficientes para estender e delimitar a curvatura Gaussiana, curvatura média e curvaturas principais perto de todos os tipos de singularidades das frentes. Além disso, estudamos a convergência para limites infinitos desses invariantes geométricos e mostramos como isso está estreitamente relacionado a uma propriedade de aproximação de frentes por superfícies paralelas
 
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Publishing Date
2021-08-27
 
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