Geometry in Minkowski space

• Grant number: 13/02794-4
• Support Opportunities: Regular Research Grants
• Start date: July 01, 2013
• End date: December 31, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ana Claudia Nabarro
• Grantee: Ana Claudia Nabarro
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The Minkowski space R_1^n is the vector space R^n with the pseudo-scalar product =-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 in R_1^n is spacelike if 0$, lightlike if =0 and timelike if <0. These spaces are used in Physics, for example, in relativity theory R_1^4 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. Our aim in one of then is to study curves in Minkowski space $\mathbb{R}_1^{3}$, concentrating our efforts in exploring what happens at points where the metric is degenerated. In the second line of research we intend to study spacelike surfaces in the de Sitter space $S^5_1$ throught invariants related to the second fundamental form including the curvature ellipse, binormal and asymptotic directions. In the third line of research we intend to study surfaces in the Minkowski space $\mathbb{R}_1^{4}$, specially where the metric is degenerated. (AU)

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

NABARRO, ANA CLAUDIA; SACRAMENTO, ANDREA DE JESUS. Focal set of curves in the Minkowski space near lightlike points. PUBLICATIONES MATHEMATICAE-DEBRECEN, v. 88, n. 3-4, p. 487-510, 2016. (10/20301-7, 13/02794-4)

IZUMIYA, SHYUICHI; NABARRO, ANA CLAUDIA; SACRAMENTO, ANDREA DE JESUS. HOROSPHERICAL AND HYPERBOLIC DUAL SURFACES OF SPACELIKE CURVES IN DE SITTER SPACE. JOURNAL OF SINGULARITIES, v. 16, p. 180-193, DEC 2017. (10/20301-7, 13/02794-4)

KASEDOU, MASAKI; NABARRO, ANA CLAUDIA; SOARES RUAS, MARIA APARECIDA. SECOND ORDER GEOMETRY OF SPACELIKE SURFACES IN DE SITTER 5-SPACE. PUBLICACIONS MATEMATIQUES, v. 59, n. 2, p. 449-477, 2015. (13/02794-4)

IZUMIYA, SHYUICHI; NABARRO, ANA CLAUDIA; SACRAMENTO, ANDREA DE JESUS. Pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz-Minkowski space. JOURNAL OF GEOMETRY AND PHYSICS, v. 97, p. 105-118, NOV 2015. (13/02794-4, 10/20301-7)