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Thèse de Doctorat
DOI
https://doi.org/10.11606/T.55.2020.tde-03022020-165513
Document
Auteur
Nom complet
Dirce Kiyomi Hayashida Mochida
Unité de l'USP
Domain de Connaissance
Date de Soutenance
Editeur
São Carlos, 1993
Directeur
Jury
Fuster, Maria Del Carmen Romero (Président)
Costa, Sueli Irene Rodrigues
Garcia, Ronaldo Alves
Loibel, Gilberto Francisco
Marar, Washington Luiz
Titre en portugais
GEOMETRIA GENÉRICA DE SUBVARIEDADES DE CODIMENSÃO MAIOR QUE 1 EM Rn
Mots-clés en portugais
Não disponível
Resumé en portugais
Não disponível
Titre en anglais
Not available
Mots-clés en anglais
Not available
Resumé en anglais
We study the geometry of m-submanifolds embedded in Rm+k, k ≥ 2, through their generic contacts with hyperplanes. Our approach to this problem is to relate the geometric properties of the submanifold to the ones of the canal hypersurface associate to it. We consider the Gauss mapping defined on the canal hypersurface and the relationship between its generic singularities and the corresponding singularities of the families of height functions defined on the manifold. We give definitions of osculating hyperplanes and binormal directions of an m-submanifold of Rm+k, and as a consequence of the setting we analyse the existence of binormal directions in terms of the generic geometry of the submanifold. We also show that for a generic embedding, the restriction to the parabolic set of the natural projection of the hypersurface to the submanifolds is a stable mapping. We use this mapping to define asymptotic directions on the submanifolds and to obtain a duality relationship between binormal directions and asymptotic directions of the submanifold. This result extend the local aspects of a result due to Bruce and Romero Fuster in [11]. The generic embeddings of surfaces in R4 are studied in great detail the inflection points of then are shown to be the umbilic points of their families of height functions. As a consequence of the generic behavior of the umbilic points of locally convex embeddings of surfaces in R4, we prove that for such embeddings, 2 | x{M) ≤ the number of inflection points of M. In particular locally convex embedded surfaces with non-zero Euler characteristic have at least 4 inflection points (this is an extension to 4-space of a result of Feldman in [21]).
 
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Date de Publication
2020-02-03
 
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