Differential Geometry of curves and surfaces in non euclidian spaces

• Grant number: 25/08260-9
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2025
• End date: June 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Alexandre Paiva Barreto
• Grantee: João Pedro Bonfim
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 22/16455-6 - Algebraic, geometric, and differential topology, AP.TEM

Abstract

Unlike traditional courses in Differential Geometry, which are restricted to the Euclidean space, in this project we are interested in studying the properties of curves and surfaces immersed in other 3-dimensional geometric environments. Due to their good geometric properties and relative simplicity, the environments chosen to begin our studies were: the spherical space and the hyperbolic space. In addition to offering a deeper understanding of the Euclidean environment itself, the study of Differential Geometry in non-Euclidean environments naturally leads the student to the Riemannian Geometry and already directs them towards the scientific research in thearea of Geometry.The ultimate goal of this project is the study of Weingarten curves and surfaces, that is, curves and surfaces whose geometric invariants (curvature, torsion, mean curvature, Gaussian curvature, principal curvatures, etc.) verify a smooth relationship. The study of such curves and surfaces is a classic topic in Differential Geometry that began in the 19th century with the work of Weingarten and remains an active research area, full of open problems and conjectures. (AU)