Differential geometry is the study of objects of all dimensions on which you can do calculus. The first great differential geometer was Gauss, and we will study his theory of surfaces.
This course focuses on 1-dimensional objects (curves) and 2-dimensional objects (surfaces), but higher dimensional manifolds are also fascinating. In order to do calculus, you have to have a good way to measure distances along paths in the object and to measure angles formed by intersecting paths (in dimension 2 and higher). In jargon, you must have a metric.
Curves and surfaces can be studied using an extrinsically defined geometry. For example, the curve or surface may be be embedded in , and the geometry of 3-space is controlled entirely by dot products of 3-dimensional vectors, which is a Riemannian geometry. Other geometries are possible, such as a Minkowski geometry that underpins the theory of relativity, or a symplectic geometry that underlays the fancy modern theory of mechanics, but we will not go there.
Curves and surfaces can also be studied intrinsically, which means that all properties must be defined and computable using only points in the curve or surface itself, without reference to a larger space (if any) in which the curve or surface resides. One of the themes of differential geometry has been to determine which features of a geometry are intrinsic and which are extrinsic. The culmination of the course will be the Gauss-Bonnet theorem relating the (intrinsic) topology of a surface to its (intrinsic) curvature.
- Curves in the plane
- Curves in space
- Surfaces in space, extrinsic geometry
- Map projections
- Surfaces in space, intrinsic geometry and the Gauss-Bonnet theorem
- Hyperbolic plane
Flath’s notes on the textbook Differential Geometry of Curves and Surfaces by Thomas Banchoff and Stephen Lovett.