**
Math 42, Winter 2006, **

Differential Geometry I
##
Covered topics

- 1.
** Curves **

- Length and parametrizations
- Curvature, torsion and the Frenet formulae
- Isoperimetric Inequality

- 2.
**Surfaces and Curvature **

- Surfaces: parametrizations and graphs
- The tangent space and vector fields
- Immersed and Embedded surfaces
- The normal vector and Guass map
- Curvatures
- Parallel transport

- 3.
** Geodesics and Isometries **

- Surfaces as metric spaces
- Geodesics
- Geodesic equations: connections and curvature
- Isometries

As time permits, some of the following topics

- Constant Mean Curvature Surfaces
- Topology and geometry: Euler characteristic and the Gauss-Bonnet Theorem
- Abstract Surfaces
- Higher dimensions