On Umbilic Points on Newly Born Surfaces

Autor(es)

Hasegawa, Masaru ^[1] ; Tari, Farid ^[2]

Afiliação do(s) autor(es)

^[1] Iwate Med Univ, Ctr Liberal Arts & Sci, Dept Informat Sci, 2-1-1 Nishitokuda, Yahaba Cho, Yahaba, Iwate 0283694 - Japan

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Ave Trabalhador Sao Carlense, 400 Ctr, BR-13566590 Sao Carlos, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY; v. 48, n. 4, p. 679-696, DEC 2017.
• Citações Web of Science: 0

Resumo

The simplest way to have birth of surfaces is through transitions in the fibres of a function f with a Morse singularity of index 0 or 3. It is natural to seek to understand the geometry of newly born surfaces. We consider here the question of finding how many umbilics are on a newly born surface. We show that newly born surfaces in the Euclidean 3-space have exactly 4 umbilic points all of type lemon, provided that the Hessian of f at the singular point has pairwise distinct eigenvalues. This is true in both cases when f is an analytic or a smooth germ. When only two of such eigenvalues are equal, the number of umbilic points is either 2, 4, 6 or 8 when f is an analytic or a generic smooth germ. The same results holds for newly born surfaces in the Minkowski 3-space. In that case when the two eigenvalues associated to the two spacelike eigenvectors are distinct we get exactly 4 umbilic points all of type lemon. If they are equal, the number of umbilic points is either 2, 4, 6 or 8. (AU)

• Processo FAPESP: 16/02701-4 - Teroria das singularidades plana e redonda
• Beneficiário: Farid Tari
• Linha de fomento: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 14/00304-2 - Singularidades de aplicações diferenciáveis: teoria e aplicações
• Beneficiário: Maria Aparecida Soares Ruas
• Linha de fomento: Auxílio à Pesquisa - Temático
• Processo FAPESP: 13/02543-1 - A geometria de superfícies singulares de ponto de vista da teoria das singularidades
• Beneficiário: Hasegawa Masaru
• Linha de fomento: Bolsas no Brasil - Pós-Doutorado