Cr-invariants for surfaces in R^4

Author(s)

Jorge Luiz Deolindo Silva

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2016-01-28

Examining board members

Farid Tari; Carlos Henrique Grossi Ferreira; Luciana de Fátima Martins; Raúl Adrián Oset Sinha; João Nivaldo Tomazella

• Advisor: Farid Tari

Abstract

In this thesis we study the extrinsic geometry of smooth surfaces in R4 via their contact with lines and hyperplanes. Uribe-Vargas introduced a cr-invariant (crossratio) at a cusp of Gauss of a surface in R3. For a surface in R4, the point P3(c) has similar behavior to that of a cusp of Gauss of a surface in R3. We establish in this thesis cross-ratio invariants for surfaces in R4 in an analogous way to Uribe- Vargass work for surfaces in R3. We study the geometric locii of local and multilocal singularities of ortogonal projections of the surface and classify the k-jets of parametrizations of germs of surfaces in the projection space P4 given in Monge form by projective transformations. The cross-ratio invariants at P3(c) points are used to recover two moduli in the 4-jet of the projective parametrization of the surfaces. (AU)

• FAPESP's process: 12/00066-9 - On the geometry of surfaces in R4
• Grantee: Jorge Luiz Deolindo Silva
• Support Opportunities: Scholarships in Brazil - Doctorate