What to give as final lecture in a differential geometry class? During the fall semester, I had to give an exercise class to second year math students, as support for a theoretical class loosely based on the book `Differential geometry of curves and surfaces' by Do Carmo. Now and then I succeeded in squeezing in some fun topics not covered in the theory class, like the four-vertex theorem, minimal surfaces, de Rham cohomology of $\mathbb{R^{3}}$, ... Next week I have to give the last class, and I want to finish with some very nice theorem or set of ideas. Does anyone have any experience with this? Maybe someone was once in a class that tried to do something similar? There are a couple of criteria:


*

*It should take about an hour to explain

*It should only need the embedded definition of a variety, not the intrinsic one

*It is around the level of Do Carmo

*The prerequisites are the first and second fundamental form, normal and Gauss curvatures, the theorema egregium and basic ideas on geodesics and geodesic curvature.


One of the physicists in the class asked me to say something about general relativity and differential geometry, but having looked through a number of references, this seems rather hard.  

The Cauchy-Crofton theorem is a nice one.  Have you not covered Gauss-Bonnet?  That's a standard. 
