Axiumbilic singular points on surfaces immersed in r4 and their generic bifurcations

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dc.creatorGarcia, Ronaldo Alves
dc.creatorTello, Jorge Manuel Sotomayor
dc.creatorSpindola, Flausino Lucas Neves
dc.date.accessioned2017-09-18T10:41:08Z
dc.date.available2017-09-18T10:41:08Z
dc.date.issued2014
dc.description.resumoHere are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little [7], Garcia - Sotomayor [11], has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic points which are the singularities of the elds of principal, mean axial lines, axial crossings and the quartic di erential equation de ning them. The Lie-Cartan vector eld suspension of the quartic di erential equation, giving a line eld tangent to the Lie-Cartan surface (in the projective bundle of the source immersed surface which quadruply covers a punctured neighborhood of the axiumbilic point) whose integral curves project regularly on the lines of axial curvature. In an appropriate Monge chart the con gurations of the generic axiumbilic points, denoted by E3, E4 and E5 in [11] [12], are obtained by studying the integral curves of the Lie-Cartan vector eld. Elementary bifurcation theory is applied to the study of the transition and elimination between the axiumbilic generic points. The two generic patterns E1 34 and E1 45 are analysed and their axial con gurations are explained in terms of their qualitative changes (bifurcations) with one parameter in the space of immersions, focusing on their close analogy with the saddlenode bifurcation for vector elds in the plane [1], [10]. This work can be regarded as a partial extension to R4 of the umbilic bifurcations in Garcia - Gutierrez - Sotomayor [5], for surfaces in R3. With less restrictive di erentiability hypotheses and distinct methodology it has points of contact with the results of Gutierrez - Gui~nez - Casta~neda [3].pt_BR
dc.identifier.citationGARCIA, R.; SOTOMAYOR, J.; SPINDOLA, F. Axiumbilic singular points on surfaces immersed in r4 and their generic bifurcations. Journal of Singularities, Massachusetts, v. 10, p. 124-146, 2014.pt_BR
dc.identifier.doi10.5427/jsing.2014.10h
dc.identifier.urihttp://repositorio.bc.ufg.br/handle/ri/12485
dc.language.isoengpt_BR
dc.publisher.countryEstados unidospt_BR
dc.publisher.departmentInstituto de Matemática e Estatística - IME (RG)pt_BR
dc.rightsAcesso Abertopt_BR
dc.titleAxiumbilic singular points on surfaces immersed in r4 and their generic bifurcationspt_BR
dc.typeArtigopt_BR

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