MATH 322
Differential Geometry
Last Offered Fall 2011
Division III Quantitative/Formal Reasoning
This course is not offered in the current catalog

It is easy to convince oneself that the shortest distance from equatorial Africa to equatorial South America is along the equator. This illustrates the fact that “straight lines” on a sphere are described by so-called great circles. It is somewhat more difficult to describe the shortest path between two points on the surface of, for example, a doughnut, reflecting the fact that a doughnut curves in space in a more complicated way than the sphere. Differential geometry is the mathematical language describing these curvature properties. In this course we will learn this language and use it to answer many interesting questions. We will also develop the tools needed to begin the more advanced study of “Riemannian” geometry, which describes (among other things) Einstein’s Relativity Theory. Topics: Curves in space, the Frenet-Serret Theorem, the first and second fundamental forms, geodesics, principal/Gaussian/mean/normal curvatures, the Theorema Egregium, the Gauss-Bonnet formula and Theorem, introduction to n-dimensional Riemannian manifolds/metrics/curvature.