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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2020.tde-03022020-165513
Document
Author
Full name
Dirce Kiyomi Hayashida Mochida
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 1993
Supervisor
Committee
Fuster, Maria Del Carmen Romero (President)
Costa, Sueli Irene Rodrigues
Garcia, Ronaldo Alves
Loibel, Gilberto Francisco
Marar, Washington Luiz
Title in Portuguese
GEOMETRIA GENÉRICA DE SUBVARIEDADES DE CODIMENSÃO MAIOR QUE 1 EM Rn
Keywords in Portuguese
Não disponível
Abstract in Portuguese
Não disponível
Title in English
Not available
Keywords in English
Not available
Abstract in English
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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Publishing Date
2020-02-03
 
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