Introduction to the differential geometry: manifolds, Riemannian geometry and fibre bundles

• Grant number: 21/02733-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): July 01, 2021
• Effective date (End): June 30, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Fernando David Marmolejo Schmidtt
• Grantee: Julia Carolina Gonçalves
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

Differential Geometry originated from the union between calculus and geometry. Over the centuries, it has shown up indispensable for the development and study of physics' theories, and it is present in the construction of formalism from Classical Mechanics and Thermodynamics, to Electromagnetism and General Relativity, the latter being inconceivable without this branch of the mathematics. Supported by these facts, this research aims to introduce the student in this field of mathematical knowledge, encompassing manifolds - reals and complexes-, Riemannian geometry and fiber bundles - from definitions to connections. Therefore, it aspires to provide the student the necessary discernment for a deeper future study in the scope of theoretical physics, since despite the importance, the teaching of this mathematics is scarce during the graduation.