Tópicos de Variedades Diferenciables: Métricas de Riemann.

Abstract

The purpose of this curricular integration work is to describe geometric properties in the context of Riemann manifolds like angles, lengths, distances, geodesics and minimizing curves. In order to achive that, we first defined and stated the properties associated with the cotangent bundle, tensor theory, and tensor fields. Next, we defined Riemannian metrics, which allowed us to introduce Riemannian manifolds. We immediately defined geometric properties such as angles, lengths and distances, and showed results related to these concepts. In the Section 2.3, we presented the definitions of geodesics and minimizing curves. Finally, in Subsections 2.3.7 and 2.3.8, we discussed and proved the connection between a geodesic and a minimizing curve.