Course Identification

The main goal of this course is to outline geometry as an area of modern mathematics, and to show that this presentation is essential, relevant and makes sense for math teachers at the high school level. Students will be accustomed to viewing concepts from more than one viewpoint.

1. Topics in trigonometry.
2. Curves in the plane - parametric representations and other representations (and their significance), tangent, normal, length, curvature, conic sections and quadratic forms.
3. Topic in linear spaces.
4. The concept of a surface as represented in several ways.
5. Elementary differential geometry: first fundamental form, tensors, second fundamental form, curvature(s), Gauss map, Gauss theorem.
6. Euclidean metric (length, area, volume), Riemannian metric, geodesics, elementary calculus of variations. Spherical geometry is going to be emphasized.
7. Topics in: Euclidean geometry (metric), various non-Euclidean geometries, analytic and synthetic approaches, a new look on conic sections and quadratic forms, different representations of the same objects, the Erlangen program.

Right from the beginning, we will emphasize the need for invariant representations of objects on one hand and easily computable representations on the other hand.

Upon successful completion of the course students will be able to:

1. Demonstrate proficiency in the different topics in the field of Geometry.
2. Recognize the relevancy of Geometry to their everyday practice as mathematics teachers in high schools.
3. Look at geometry concepts from more than one viewpoint.

do Carmo, Manfredo P., Differential Geometry of Curves and Surfaces

Wikipedia entries on subject discussed and the links there.