Teorema de Gauss-Bonnet via formas dferenciais

Abstract

This monograph focuses on demonstrating and understanding the geometric motivation of the Gauss-Bonnet Theorem. To this end, the project initially envisages the study of differentiable manifolds, tangent space, cotangent space, surface orientation and Frobenius’ theorem. Sequentially, the study follows the study of tensor and vector bundles, connections in vector bundles, affine connections and connections in reference bundles. The project ends with the study of Riemannian geometry, which will include the fundamental theorem of Riemannian geometry, geodesic normal coordinates, sectional curvature and, finally, the Gauss-Bonnet theorem.