Curves in Minkowski space

• Grant number: 10/20301-7
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: May 01, 2011
• End date: February 28, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ana Claudia Nabarro
• Grantee: Andrea de Jesus Sacramento
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 08/54222-6 - Singularities, geometry and differential equations, AP.TEM

Abstract

The problems for the thesis are part of a larger project that study the geometry of submanifolds in Minkowski space by using singularity theory. For her thesis, Andrea will study the geometry of curves in 3-Minkowski space. It may happen that the normal vector and binormal vector of the curve are lightlike, as can also happen that the normal plane or osculating plane is lightlike at some points on the curve. These points have not the Serret-Frenet formulas for curves. Also the curvature and/or torsion of the curve at these points are not defined. However, the contacts of the curve with planes and pseudo-spheres are well defined. Andrea will study families of height functions and distance squared functions over the curve to obtain geometric informations about the curve, including informations about the dual and focal set of the curves. We will also work with curves in timelike surfaces in Minkowski space and study the focal set of these curves and also obtain geometric results. We hope to find some invariants using the theory of contact. If there's enough time, the student will study properties of closed curves in 3-Minkowski space and will investigate the geometry of surfaces generated by curves. (AU)

Scientific publications

(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)

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)

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)

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)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

SACRAMENTO, Andrea de Jesus. Curves in the Minkowski space. 2015. Doctoral Thesis - Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) São Carlos.