Introduction to Differential and Riemannian Geometry

• Grant number: 24/07647-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2024
• End date: July 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Eduardo Rosinato Longa
• Grantee: Victor Hugo dos Santos Lins
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

From the study of differentiable manifolds, central objects in differential geometry, the student will develop the necessary tools to prove Stokes' theorem on manifolds-a statement about the integration of differential forms-obtaining the important corollaries that follow. In the second half of the project, the intention is to enter and become familiar with Riemannian geometry, aiming to explore different notions of curvature, particularly Ricci curvature. Finally, the student will apply the acquired knowledge to demonstrate the important theorem of Lichnerowicz-Obata, which establishes an optimal estimate for the first positive eigenvalue of the Laplacian on a closed Riemannian manifold.