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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.

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Does it have to be rigorous? You could show them some nice results of Morse theory. Or use de Rham cohomology to prove the Jordan curve theorem. –  Ragib Zaman Dec 17 '12 at 9:15
    
It doesn't have to be completely rigorous, but I definitely want them to get a feel for the proofs. What kind of results are you thinking of? –  theo Dec 17 '12 at 14:49
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2 Answers

From Gauss-Bonnet one quickly gets the Poincare-Hopf Theorem, which is an important and interesting result in its own right. You also get the corollary that you can't comb the hair on a compact oriented surface unless its homeomorphic to a torus (the Hairy Ball theorem in the case of a sphere). Now you have the tools to prove the following theorem of Morse:

Let $f :\Sigma\to \mathbb{R}$ be a smooth function on a compact oriented surface such that all the critical points of f are nondegenerate. Let M = number of local maxima, m= number of local minima, and s= number of saddle points. Then $$ M-s+m = \chi (\Sigma).$$

Impress on them that this is quite remarkable - it doesn't depend on what $f$ actually is. The topology of the surface alone determines in some sense how extrema form and deform. As a simple application, you can take the height function on a compact oriented surface with $g$ holes to see that it has Euler characteristic $2-2g.$

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

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I'd like to do something on surfaces, the Cauchy-Crofton theorem is definitely nice, but I already discussed related results when I discussed curves. Gauss-Bonnet is covered. –  theo Dec 17 '12 at 14:48
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