This course is intended for science majors who need to have knowledge about the geometry of curves and surfaces in space and want to have an introduction to manifolds. It is specially appropiate for math, physics and astronomy majors.
Differential Geometry of curves and surfaces, Manfredo P. do Carmo, 1976. Elementary Topics in Differential Geometry, John A. Thorpe, 1979.
Although the content of this course might change with the instructor, usually the course will be focused on giving the student hands-on experience in the treatment and description of surfaces, while introducing basic concepts such as regularity, fundamental forms, Gauss map, vector fields, covariant derivatives, geodesics and more. Depending on the instructor the study of surfaces might be substituted by the study of n dimensional hypersurfaces in the Real n+1 dimensional space. The understanding of these concepts will prepare the student for the study of differentiable manifolds and Riemannian Geometry. This is the language in which many subjects are formulated, including General Relativity and many other branches of Mathematical Physics.
This course prepares the student for a first graduate course in Differential or Riemannian geometry.
- Review of the geometry of curves
- Regular surfaces
- First fundamental form
- Second fundamental form and the Gauss map
- Vector fields
- Minimal surfaces
- Gauss Theorem and equations of compatibility
- Parallel transport, Geodesics and Gauss Bonet
- The Exponential map
- Some concepts from Global differential Geometry, as time allows