A transformação vetorial de Ribaucour para subvariedades de curvatura constante

• Autor(a) principal: Guimarães, Daniel da Silveira
• Data de Publicação: 2015
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Repositório Institucional da UFSCAR
• Texto Completo: https://repositorio.ufscar.br/handle/ufscar/7583
• topic: Geometria diferencial; Transformação de Ribaucour; Espaços de curvatura constante; Subvariedade Lagrangiana; Imersão isométrica horizontal; CIENCIAS EXATAS E DA TERRA::MATEMATICA; CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA

In this work we obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds with constant sectional curvature of space forms. As a consequence, a process is derived to generate a new family of such submanifolds starting from a given one. We prove a decomposition theorem for this transformation, from which the classical permutability theorem for the Ribaucour transformation of submanifolds with constant sectional curvature follows. Given k scalar Ribaucour transforms of a submanifold with constant sectional curvature, we prove the existence of a Bianchi k-cube all of whose vertices are submanifolds with the same constant sectional curvature, each of which is given by means of explicit algebraic formulas. A further reduction of the transformation is shown to preserve the class of Lagrangian submanifolds of dimension n and constant sectional curvature c of complex space forms of complex dimension n and constant holomorphic sectional curvature 4c. In particular, explicit parametrizations in terms of elementary functions of examples with arbitrary dimension and curvature are provided. A decomposition theorem and a version of the Bianchi cube for this transformation are also obtained.