This is a compulsory lecture module for MMath students in their fourth year.
The module will begin by looking at differential manifolds and the
differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. The module will then look at calculus on manifolds including the study of vector fields, tensor fields and the Lie derivative.
The module will then go on to study Riemannian geometry in general by showing how the metric may be used to define geodesics and parallel transport, which in turn may be used to define the curvature of a Riemannian manifold.
As another major application the module will investigate groups, such as the rotation group SO(3), which also have the structure of a manifold. Such objects are called Lie groups and play an important role in both theory and application of geometry. As an example of this we look at the symmetries of Riemannian manifolds. The isometries of a Riemannian metric form a group and the corresponding infinitesimal isometries form a Lie algebra. Another class of examples will be provided by matrix groups.