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So our instructor wants us to write a 3 page report on some fact in Differential Geometry relating to curves or surfaces. It can be a theorem, a fact or some special case.

The only restriction is that I cannot write about things we did in class. We are using a book called Elementary Differential Geoemetry by Andrew Presley. We've covered pretty much all of the book.

This includes main theorems like Gauss Bonnet, Theorema Egregium, Gaussian/Mean curvatures, Euler's Theorem, Geodesic, Gauss Map, the First/Second Fundamentals, just scratched on the surface of Minimal surfaces.

After googling some topics, I was overwhelmed due to the shocking level of mathematics involved and have thus no where else to look. So if anyone could suggest some topics for me, it would be awesome. Preferably keep it to the level of someone who only have had a course in calculus of severable variables and linear algebra. But if the restrictions are too much, you may assume that the student have had a course in Real Analysis.

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If you didn't cover it, one result from the theory of differential forms you might want to learn a little about is the generalized Stokes' theorem, of which the divergence theorem, Green's theorem, Stokes' theorem, the fundamental theorem of calculus, and other stuff like Archimedes' Principle are all extremely easy special cases. I would highly recommend Spivak's Calculus on Manifolds, a cute book of around 150 pages (almost all of which you already know) to learn the very basics.

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I am not sure if the instructor would include those "calculus theorems" as part of Diff Geometry. I have Spivak and Munkre's book –  mathet Mar 24 '13 at 2:52
    
Actually we barely even talked about what a manifold is, we can't go too deep –  mathet Mar 24 '13 at 3:02
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Poke around in Ted's book http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf and see if you can find a topic that you like. Let me also suggest some topics to you that I would guess you might not have covered in class.

1) You could talk about different models of the hyperbolic plane, e.g. the Poincare Disk Model or the upper half plane model. You could discuss the law of sines/cosines in hyperbolic space and the hyperbolic pythagorean theorem.

2) Same thing with spherical geometry. Talk about the law of sines/cosines in a sphere, talk about the geodesics, etc..

3) Pick your favorite minimal surface and discuss it's geometry. You could choose the catenoid, enneper's minimal surface, etc.. Calculate the curvature, second fundamental form, draw a bunch of geodesics, see if you can make a pythagorean theorem.

4) Show that every compact embedded surface in $\mathbb R^3$ has a point of positive Gauss curvature (see Ted' book).

5) Prove Hadamard's theorem: Any isometric immersion of a compact surface with positive Gauss curvature must be an isometric embedding. (For this you obviously need to learn what an immersion and an embedding is).

6) Prove the following version of Herglotz theorem: The image of any isometric immersion of the round metric on the 2 sphere must be a sphere in Euclidean space.

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