MATH 458. Honours Differential Geometry.

Local theory of plane and space curves: the Frenet formulas. Local theory of surfaces: the first and second fundamental forms, the shape operator, the mean and Gaussian curvatures, the Theorema Egregium, surfaces of revolution with prescribed curvature, ruled and developable surfaces, minimal surfaces. Geodesic curves on surfaces of revolution. The Gauss-Codazzi equations, rigidity. Selected topics in the global theory of plane and space curves, and in the global theory of surfaces: total curvature and the Fary-Milnor theorem on knotted curves, abstract surfaces as 2-d manifolds, the Euler characteristic, and the Gauss-Bonnet theorem for surfaces.