Next: Geometry, topology and homotopy

# Differential Geometry

C.T.J. Dodson

General description: Differential geometry begins with the study of curves and surfaces in three-dimensional Euclidean space. Using vector calculus and moving frames of reference on curves embedded in surfaces we can define quantities such as Gaussian curvature that allow us to distinguish among surfaces. The extension of the theory to higher dimensions leads to the study of n-manifolds, 1-manifolds being curves and 2-manifolds being surfaces. A distance structure was provided by Riemann and so we study Riemannian n-manifolds, their curvature and curves in them. We consider geodesic (literally, `divide the Earth') curves, which give extremal paths in n-manifolds, generalizing lines in the plane and great circles on a sphere.
Some animations of curves, surfaces and knots
Famous curves
Famous surfaces

Notes On Making Mathematical Notes For Your Course (PDF format).

Course description: Tangent vectors, vector fields, differentiable maps; curves, Frenet frames; surfaces, shape operator, Gaussian and mean curvature; n-manifolds, metric, Riemann curvature and geodesics.
See the online documents relating to this course: 351 Background Notes , Curves , Surfaces
Manifolds
Tutorial Problems 1 Tutorial Problems 2 Tutorial Problems 3 Tutorial Problems 4

Mathematica Notebooks on curves and surfaces from 117----To use, download and open in Mathematica
Introductory Mathematica
Plane curves
Space curves
Surfaces

Research in geometry: If you are interested in graduate work in an area of geometry, then please contact me in person or by email: dodson@umist.ac.uk
Other On-Line Mathematical Materials:

Kit Dodson 2002-09-08