Partially umbilic singularities of hypersurfaces of R-4

Autor(es)

Lopes, D. ^[1] ; Sotomayor, J. ^[2] ; Garcia, R. ^[3]

Afiliação do(s) autor(es)

^[1] Univ Fed Sergipe, Dept Matemat, BR-49100000 Sao Cristovao, SE - Brazil

^[2] Univ Sao Paulo, Inst Matemat & Estat, BR-05508090 Sao Paulo, SP - Brazil

^[3] Univ Fed Goias, Inst Matemat & Estat, BR-74001970 Goiania, Go - Brazil

• Tipo de documento: Artigo Científico
• Fonte: BULLETIN DES SCIENCES MATHEMATIQUES; v. 139, n. 4, p. 431-472, JUN 2015.
• Citações Web of Science: 1

Resumo

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in R-4 in a neighborhood of the set S of its principal curvature singularities, consisting of the points at which at least two principal curvatures are equal. Under generic conditions defined by appropriate transversality hypotheses it is proved that S is the union of regular smooth curves S-12 and S-23, consisting of partially umbilic points, where only two principal curvatures coincide. This curve is partitioned into regular arcs consisting of points of Darbouxian types D-1, D-2, D-3, with common boundary at isolated semi-Darbouxian transition points of types D-12 and D-23. The stratified structure of the partially umbilic separatrix surfaces, consisting of the boundary of the set of points through which the principal lines approach 8, established in this work, extends to hypersurfaces in R-4 the results of Darboux in {[}1] for umbilic points on analytic surfaces in R-3, reformulated by Gutierrez and Sotomayor in {[}8], to describe the umbilic separatrix structures of the umbilic types D-1, D-2, D-3, and further developed by Garcia, Gutierrez and Sotomayor in {[}6], for their D-12 and D-23 generic bifurcations. This work complements results of Garcia {[}5] on the structure of principal curvature lines around the generic partially umbilic points of hypersurfaces in R-4. (C) 2014 Elsevier Masson SAS. All rights reserved. (AU)

• Processo FAPESP: 08/02841-4 - Teoria topológica, geométrica e ergódica dos sistemas dinâmicos
• Beneficiário: Jorge Manuel Sotomayor Tello
• Modalidade de apoio: Auxílio à Pesquisa - Temático