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Geometry of manifolds in the euclidian space and in the Minkowski space

Grant number: 16/19139-7
Support type:Regular Research Grants
Duration: November 01, 2016 - April 30, 2019
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Ana Claudia Nabarro
Grantee:Ana Claudia Nabarro
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The Minkowski space $\mathbb{R}_1^{n}$ is the vector space Rn with the pseudo-scalar productu.u=-u_1v_1+...+u_nv_n$, for any u=(u_1,...,u_n) and v =(v_1,..., v_n) inR_1^{n}. A non zero vector u em R^n_1 is spacelike if u.u>0$, lightlike ifu.u=0$ and timelike if u.u<0. These spaces are used in Physics, for example, in relativity theory R14 is a model for space and time. The induced metric from the above scalar product is an example of a Lorentzian metric and is called the Minkowski metric.Many studies have been carried out in these spaces and interesting results have been proved.In some ways, the Lorentzian geometry complements the Riemannian geometry. Challenging problems arrise when the induced metricon a submanifold in the Minkowski space changes signature. It is interesting, for example, to study what happensat points where the metric is degenerate and explain the changes in the geometry, say, from a Riemannian region to aLorentzian region of the submanifold.This project has three lines of research in the Minkowski space. We will use singularity theory to get geometric informations about the submanifolds. In a joint work with professor Shyuchi Izumiya of Hokkaido University and with Andrea de Jesus Sacramento, we want to study curves on spacelike and timelike hypersurfaces in $\mathbb{R}_1^{4}$. With Maria Aparecida Soares Ruas (ICMC) and Masaki Kasedo of Akita National College of Technology, Japan, we will study asymptotic lines on spacelike surfaces in the de Sitter space $S^5_1$ and their singular points. In a joint work with Farid Tari (ICMC) and Federico Sanches-Bringas of National Autonomous University of México, we are studying surfaces in the Minkowski space $\mathbb{R}_1^4$ specially where the metric is degenerated. In the Euclidian space we will study the geometry of hypersurfaces in $\mathbb{R}^4, $ also using singularity theory. Our aim is to study the singularities of the Gauss map of families of hypersurfaces in $\mathbb{R}^4$. This is a joint work with Maria Carolina Zanardo (PhD student of ICMC) and Maria del Carmen Romero-Fuster of the University of Valencia. Besides, with M.C. Romero-Fuster we want to study Lagrangian singularities and with Masaki Kasedo and Maria Aparecida Soares Ruas we intend to study vector fields in $\mathbb{R}^3$. (AU)