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Explore a comprehensive lecture on Differential Geometry of Curves and Surfaces in this 18th session of a master's level course. Delve into advanced topics such as curves in plane 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. Investigate the Gauss normal map, second fundamental form, principal curvatures, Gaussian curvature, mean curvature, curvature lines, and asymptotic lines. Analyze the sign of Gaussian curvature, mean curvature and area, the sphere rigidity theorem, and Alexandrov's Theorem. Compare intrinsic and extrinsic geometry, explore isometries and Gauss's Theorema Egregium. Learn about covariant derivatives, parallel transport, geodesic curvature, and geodesics. Discover the Gauss-Bonnet Theorem and its applications, along with other global theorems and related topics. Prerequisites include familiarity with the inverse function theorem in Rn and basic topology of Rn space.
