Geometria Diferencial de Curvas e Superfícies - Aula 14

Explore a comprehensive graduate-level lecture on Differential Geometry of Curves and Surfaces, delivered by Professor Lucas Ambrozio as part of the 2023 Summer School at the Instituto de Matemática Pura e Aplicada. Delve into advanced topics including curves in plane and space, length, curvature, torsion, surfaces in R3, parameterizations, coordinate changes, tangent planes, fundamental forms, Gaussian and mean curvatures, the Gauss normal map, principal curvatures, asymptotic lines, and intrinsic vs. extrinsic geometry. Examine key theorems such as the sphere rigidity theorem, Alexandrov's theorem, Gauss's Theorema Egregium, and the Gauss-Bonnet theorem. Prerequisite knowledge of the inverse function theorem in Rn and basic topology of Rn space is recommended for this in-depth, 2-hour and 9-minute lecture, which is part of a broader series on differential geometry.