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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Inflection points and topology of surfaces in 4-space


Authors: Ronaldo Alves Garcia, Dirce Kiyomi Hayashida Mochida, Maria del Carmen Romero Fuster and Maria Aparecida Soares Ruas
Journal: Trans. Amer. Math. Soc. 352 (2000), 3029-3043
MSC (1991): Primary 58C27; Secondary 53A05
Published electronically: March 15, 2000
MathSciNet review: 1638242
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Abstract:

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.


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Additional Information

Ronaldo Alves Garcia
Affiliation: Instituto de Matemática e Estatistica, Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil
Email: ragarcia@mat.ufg.br

Dirce Kiyomi Hayashida Mochida
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, 13560-905, São Carlos, SP, Brazil
Email: dirce@dm.ufscar.br

Maria del Carmen Romero Fuster
Affiliation: Departamento de Geometria e Topologia, Universidad de Valencia, 46000, Valencia, Spain
Email: romero@uv.es

Maria Aparecida Soares Ruas
Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Departamento de Matemática, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
Email: maasruas@icmsc.sc.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02404-1
PII: S 0002-9947(00)02404-1
Keywords: Inflection point, height function, asymptotic direction
Received by editor(s): September 29, 1997
Received by editor(s) in revised form: June 15, 1998
Published electronically: March 15, 2000
Additional Notes: Research of the first author was partially supported by CNPq and FUNAPE, Brazil.
Research of the third author was partially supported by DGCYT, grant no. PB96-0785
Research of the fourth author was partially supported by CNPq, Brazil, grant # 300066/88-0.
Article copyright: © Copyright 2000 American Mathematical Society