Various areas of interest and research within the field are described below, and the courses regularly offered in each area are listed. Many of the courses are given every year, while the rest are given whenever the demand is great enough. In addition, there are special topics courses each semester on subjects not covered by the regular courses. Examples of such structures are Riemannian manifolds and homogeneous spaces. These generalize the classical geometries of Euclid, Lobatchevski, and Riemann's spherical geometry.
Further examples are foliations.
In a Riemannian manifold a neighborhood of each point is given a Euclidean structure to a first order approximation. Classical differential geometry considers the second order effects of such a structure locally, that is, on an arbitrarily small piece. In a homogeneous space there is a distinguished group of differentiable mappings of the space into itself which acts transitively on points. An abstract space of such mappings is the prototype of a Lie group, so Math 427 and 408 are basic to their study. More advanced work in homogeneous spaces usually comes in conjunction with the other geometric structures alluded to above.