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

(1) Differential Geometry of Curves & Surfaces, Manfredo P. Do Carmo, 2nd Edition, Dover Publications
(2) Elementary Differential Geometry, Andrew Pressley, Springer
(3) An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby, Academic Press
(4) Riemannian Geometry, Manfredo P. Do Carmo, Springer

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.