Título: | ASYMPTOTIC NETS WITH CONSTANT AFFINE MEAN CURVATURE | ||||||||||||
Autor: |
ANDERSON REIS DE VARGAS |
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Colaborador(es): |
MARCOS CRAIZER - Orientador |
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Catalogação: | 26/AGO/2021 | Língua(s): | ENGLISH - UNITED STATES |
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Tipo: | TEXT | Subtipo: | THESIS | ||||||||||
Notas: |
[pt] Todos os dados constantes dos documentos são de inteira responsabilidade de seus autores. Os dados utilizados nas descrições dos documentos estão em conformidade com os sistemas da administração da PUC-Rio. [en] All data contained in the documents are the sole responsibility of the authors. The data used in the descriptions of the documents are in conformity with the systems of the administration of PUC-Rio. |
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Referência(s): |
[pt] https://www.maxwell.vrac.puc-rio.br/projetosEspeciais/ETDs/consultas/conteudo.php?strSecao=resultado&nrSeq=54401&idi=1 [en] https://www.maxwell.vrac.puc-rio.br/projetosEspeciais/ETDs/consultas/conteudo.php?strSecao=resultado&nrSeq=54401&idi=2 |
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DOI: | https://doi.org/10.17771/PUCRio.acad.54401 | ||||||||||||
Resumo: | |||||||||||||
Discrete Differential Geometry aims to develop a discrete theory which respects fundamental aspects of smooth theory. With this in mind, some results of smooth theory of Affine Geometry are firstly introduced since their discrete counterparts shall be treated a posteriori. The first goal of this work is construct a discrete affine structure for nets in a three-dimensional space with indefinite Blaschke metric and asymptotic parameters. For this purpose, one defines a conormal vector field, which satisfies
Lelieuvre s equations and it is associated to a real parameter; and an affine normal
vector field, which defines the cubic form of the net and makes the structure well
defined. This structure allows to study, e.g., ruled surfaces with emphasis on improper
affine spheres. Moreover, a definition for singularities is proposed in the case of discrete
improper affine spheres from the center-chord construction. Another goal here is to
propose a definition for an asymptotic net with constant affine mean curvature
(CAMC), in a way that encompasses discrete affine minimal surfaces and discrete affine
spheres. Discrete affine minimal surfaces receive a beautiful geometrical
characterization directly linked to discrete Lie quadrics. This work is completed with
the main result about a discrete version of Cayley surfaces, which are ruled improper
affine spheres that can be characterized by the induced connection as: an asymptotic net
with CAMC is equiaffinely congruent to a Cayley surface if and only if the cubic form
does not vanish and the affine induced connection is parallel.
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