Spherical and Hyperbolic Geometries with applications to differential geometry

• Grant number: 21/02517-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: June 01, 2021
• End date: December 31, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Marta Cilene Gadotti
• Grantee: Maria Júlia Fassis
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

With this project, we intend to introduce the essential elements of Geometry and Hyperbolic Trigonometry and also Spherical Geometry and study results that involve these elements, especially those applied to Differential Geometry. Let's go first carry out the study of the upper semiplane and the Poincaré disk. Next we will approach the transformations that keep the disk and the invariant semiplane, the Transformadogs of M obius and the study of Isometries. We also want to address some trigonometric relations and we will explain about the Riemann surface. Furthermore, in a broader context, we will attack some topics of differential geometry.