Geometry of Curves and Surfaces

Week 1-3: Curves in R² and R³, Regular Curves, Arc Length, Local theory of Curves: Curvature & torsion of a curve, Frenet-Serret Formulas, Fundamental Theorem of the local theory of curves, Global properties of plane curves : Isoperimetric inequality and the four vertex theorem.
Week 4-7:Review of multivariable calculus : Differentiable functions, Implicit function theorem, Inverse function theorem. Regular Surfaces in R³, Regular values of a differentiable function, Inverse image of regular values, Differentiable functions on surfaces, Tangent Plane on surfaces, Differential of a Map, First Fundamental Form, Length of curves on surfaces, Area, orientation on a surface, orientable surfaces.
Week 8-10:Gauss Map and its Fundamental Properties, Second Fundamental Form, Normal Curvature, Principal Curvature, Gaussian Curvature, Classification of point on surfaces : Elliptic, Hyperbolic, Parabolic, Planar, Gauss Map in terms of Local Coordinates
Week 11&12:Isometries and local isometries between surfaces, Christoffel symbols, Gauss’s Theorema Egregium (Remarkable Theorem), Parallel Transport, Geodesics on surfaces, Triangulation of a surface, Euler Characteristic of a surface, Gauss-Bonnet Theorem