A new approach to the differential geometry of frontals in the Euclidean space

• Ano de defesa: 2021
• Autor(a) principal: Tejeda, Tito Alexandro Medina
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Curvatura Gaussiana; Curvatura Média; Curvaturas Principais; Frente; Front; Frontal; Gaussian curvature; Mean curvature; Principal curvatures
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-27082021-141339/

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.