Geometria Diferencial de Curvas e Superfícies - Aula 01

Explore the fundamental concepts of Differential Geometry of Curves and Surfaces in this comprehensive graduate-level lecture from the 2023 Summer School at the Institute of Pure and Applied Mathematics. Delve into topics such as curves in the plane and space, length, curvature, and torsion, as well as surfaces in R3, including parameterizations, coordinate changes, and tangent planes. Examine the first fundamental form, distances, areas, curvature, and the Gauss normal map. Investigate principal curvatures, Gaussian and mean curvatures, curvature lines, and asymptotic lines. Learn about the sign of Gaussian curvature, mean curvature and area, the sphere rigidity theorem, and Alexandrov's Theorem. Discover the differences between intrinsic and extrinsic geometry, isometries, and Gauss's Theorema Egregium. Study covariant derivatives, parallel transport, geodesic curvature, and geodesics. Explore the Gauss-Bonnet Theorem and its applications, along with other global theorems and related topics.