Geometria Diferencial de Curvas e Superfícies - Aula 08

Explore a comprehensive lecture on Differential Geometry of Curves and Surfaces in this eighth installment of a master's level course taught by Professor Lucas Ambrozio. Delve into advanced topics such as curves in plane and space, length, curvature, torsion, surfaces in R3, parameterizations, coordinate changes, and tangent planes. Examine the first fundamental form, distances, areas, curvature, Gauss normal map, second fundamental form, principal curvatures, Gaussian and mean curvatures, curvature lines, and asymptotic lines. Investigate the sign of Gaussian curvature, mean curvature and area, sphere rigidity theorem, and Alexandrov's Theorem. Contrast intrinsic and extrinsic geometry, study isometries and Gauss's Theorema Egregium, covariant derivatives, parallel transport, geodesic curvature, and geodesics. Explore the Gauss-Bonnet Theorem and its applications, along with other global theorems and related topics in this intensive 1 hour 59 minute session from the 2023 Summer School at the Institute of Pure and Applied Mathematics (IMPA).