Ideas for a Project on Differential Geometry Currently trying to find a topic for a roughly fifteen page paper on Differential Geometry with presentation, with the rough target being a second year graduate student audience.
I was looking in particular for some interesting problem that can be efficiently solved using differential-geometric techniques, or at least something suitably categorical. Any ideas? I was considering discussing categorical generalizations of Lie algebras.
 A: The four-vertex-theorem or brachistochrone curves.
A: You could make an introduction to Morse theory finishing with the Reeb theorem, which states that a compact manifold admitting a Morse function with only two critical points is homeomorphic to a sphere. This would be the first three sections plus theorem 4.1 of Milnor's book. If that were too short, you can go further in any of the multiple directions Morse theory has developed.
A: How about the stories of world discoverers in the fifteenth centuries era, why lots of them did not end up where they though they would be. What is the definition of "straight" on a sphere? 
Relate it to the world which is not flat. Put it together with loxodromes (you should know what that is!) and find a mathematical mechanism that allows you to set up a path with given starting coordinates on the sphere and desired end coordinates; path having a fixed angle with due north (they only had compass remember?). There comes in your Diff Geometry theory.
Lots of path from the past were rhumb lines, but that had its own problems.
And why were loxodromes easier to follow than great circles? This is something you can explore with Diff geometry and relate to your audience  
