Cálculo de Áreas com Métricas Riemannianas

Abstract

This work provides an accessible introduction to the concept of a Riemannian metric in domains of Euclidean space, with an emphasis on its application to area calculation. Using fundamentals of Linear Algebra and Multivariable Calculus, the goal is to make the topic comprehensible for beginners, even if, at times, the matical rigor is softened in favor of intuition. The objective is to present fundamental results clearly, highlighting the elegance of the theorems and techniques involved. The text discusses the calculation of areas using the Riemannian metric, starting from the Change of Variables Theorem for integrals. Metrics in different contexts are explored, including Euclidean and hyperbolic spaces, with concrete pplications such as the computation of the area of the torus and the spheres S 2 (R) and S 3 (R). The study extends to hyperbolic space, analyzing area measurement in specific regions, such as rectangles and circular annuli, providing a theoretical and practical framework for more advanced investigations in Riemannian Geometry.