Part I : Curves :
- Curves : Parametrized and regular curves, arc length, parametrization by arc length.
- Local Theory : Tangent-Normal-Binormal frame, curvature, torsion, fundamental theorems of local theory of plane and space curves.
- Global Theory : Simple curves, the Jordan curve theorem (without proof), isoperimetric inequality, the four-vertex theorem.
Part II : Surfaces :
- Surfaces : Parametrization, change of parameters, smooth functions, tangent plane, differential, diffeomorphism, inverse and implicit function theorems.
- Second Fundamental Form and Curvature : Gauss map, oriented surfaces, second fundamental form, Gauss, mean and principal curvatures, normal sections.
- Integration on Surface : Definition of integral, partitions of unity, change of variables formula, divergence theorem.
- Global Extrinsic Geometry : Positively curved surfaces, Minkowski formulas, Aleksandrov theorem, isoperimetric inequality.
- Intrinsic Geometry : Rigid motions and isometries, Gausss theorema egregium, geodesics, existence and uniqueness of geodesics, Hopf-Rinows theorem.
- Gauss-Bonnet theorem : Degree of maps between surfaces, index of a vector field at an isolated zero, Gauss-Bonnet formula, Euler characteristic.
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