Differential Geometry gives us lots of opportunity for further study on special topics. Depending on which grade you are aiming for, you will complete projects of various types that allow you to go deeper into special parts of the subject.
A Literature Research Project
Choose an article from an expository journal which addresses some topic in classical differential geometry of curves or surfaces. Appropriate journals include:
One could also choose a topic not on our study list, but that appears in a standard reference work in differential geometry. Then write your understanding of the work for a different audience.
A Creative Project
Make something that displays an idea we have studied somehow. Ideas for things one could make:
There are lots of reasonable things one could do here. I’ll just make a short list to help you get started looking for ideas.
Bertrand Mates, Which way did the bicycle go?, contact between curves, involutes and evolutes, the isoperimetric problem, spherical images, Fenchel and Fáry-Milnor theorems, other connections to knot theory, the Cauchy-Crofton formula, Hopf’s Umlaufsatz, Ovals and the four vertex theorem, planes curves of constant width, the Darboux vector, pedal curves, the Bishop framing.
The Gauss-Bonnet Theorem, Minimal surfaces, calculus of variations and geodesics, The Hopf-Rinow theorem, the geodesic flow, ruled surfaces, Hilbert’s theorem, Liebmann’s theorem, Minding’s theorem, complete surfaces of constant curvature, surfaces of constant width, Jacobi’s last geometric “theorem”, the Beltrami-Enneper theorem, Clairaut’s theorem, Mirrors, Maps, the Rodriguez formula and Bonnet’s theorem on lines of curvature
There are so many topics, that you can easily find a whole bunch more. This is just a list I made up in an afternoon of thinking.