Differential geometry (Advanced)

This unit will explore the metric structure of curves and surfaces, primarily in 3-dimensional Euclidean space. The major focus is on the various concepts of curvature and related notions, and the relationships between them. Curvature and torsion of a curve. First and second fundamental forms of a surface. Geodesic and normal curvatures of a curve on a surface. Gaussian, mean and principal curvatures of a surface. Important theorems relating these concepts and the proofs and ideas underlying them. Links will be drawn with many other areas of mathematics, including real and complex analysis, linear algebra, differential equations, and general relativity.