GEOMETRIA GENÉRICA DE SUBVARIEDADES DE CODIMENSÃO MAIOR QUE 1 EM Rn

Mochida, Dirce Kiyomi Hayashida

doi:10.11606/T.55.2020.tde-03022020-165513

Home

Facilities

Doctoral Thesis

DOI

https://doi.org/10.11606/T.55.2020.tde-03022020-165513

Document

Doctoral Thesis

Author

Mochida, Dirce Kiyomi Hayashida (Catálogo USP)

Full name

Dirce Kiyomi Hayashida Mochida

Institute/School/College

Instituto de Ciências Matemáticas e de Computação

Knowledge Area

Geometry and Topology

Date of Defense

1993-09-09

Published

São Carlos, 1993

Supervisor

Mendes, Claudio Martins (Catálogo USP)

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 R^{m+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 R^m+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]).}

WARNING - Viewing this document is conditioned on your acceptance of the following terms of use:
This document is only for private use for research and teaching activities. Reproduction for commercial use is forbidden. This rights cover the whole data about this document as well as its contents. Any uses or copies of this document in whole or in part must include the author's name.

DirceKiyomiHayashidaMochida_DO.pdf (25.45 Mbytes)

Publishing Date

2020-02-03

Derived works

WARNING: Learn what derived works are clicking here.