The course begins with a study of curves and how they bend and twist in space. It then considers surfaces, studying the classical fundamental forms introduced by Gauss, the various measures of curvature for surfaces and what they mean for the internal and external appearance and properties of surfaces. A closer look at the intrinsic geometry of surfaces leads to Gauss' famous "Remarkable Theorem" on curvature and provides the starting point that would lead to the fundamental uses of differential geometry in, for example, Einstein's general relativity. In relation to surfaces, the course also covers geodesics, the Gauss-Bonnet theorem and the Euler characteristic. This leads to a consideration of non-Euclidean geometries, especially the hyperbolic plane.

Note: Students wishing to enrol in Level III Higher Pure Mathematics courses should consult with the Pure Mathematics Department before enrolling.