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Explore a comprehensive lecture on Differential Geometry of Curves and Surfaces in this 13th session of the Summer School 2023 master's course, delivered by Professor Lucas Ambrozio. Delve into advanced topics such as curves in planes and space, length, curvature, and torsion. Examine surfaces in R3, including parameterizations, coordinate changes, and tangent planes. Study the first fundamental form, distances, areas, and curvature concepts like Gaussian and mean curvature. Investigate the Gauss normal map, second fundamental form, principal curvatures, curvature lines, and asymptotic lines. Analyze intrinsic versus extrinsic geometry, isometries, and Gauss's Theorema Egregium. Explore covariant derivatives, parallel transport, geodesic curvature, and geodesics. Discover the Gauss-Bonnet Theorem and its applications, along with other global theorems and related topics. Prerequisite knowledge includes familiarity with the inverse function theorem in Rn and basic topology of Rn space.
