What can I do with Gauss Bonnet Theorem ( Undergraduate Project ) I have selected Gauss-Bonnet Theorem as my undergraduate project topic ( suggested by the lecturer ). I have briefly gone through the book by do Carmo until the Gauss-Bonnet Theorem but I still have about 3 months before I had to start to write my report on this topic. 
Is there any particular suggestion (book, paper) that I can further my study related to this theorem? (For example, its extension or its special case, etc.)
 A: Here's a neat extension: suppose you have a compact surface $M$ and you define $M_{\epsilon}$ to be the "inflation" of $M$ by a distance $\epsilon$ in the surface normal direction. You can show that the volume $V(M_{\epsilon})$ enclosed by the inflated surface can be related to geometric quantities on the original, with formula
$$V(M_{\epsilon}) = V(M) + \epsilon A(M) + \epsilon^2 \int_M H^2\,dA + \frac{\epsilon^3}{3}\int_M K\,dA,$$
where $A(M)$ is the surface area and $H$ and $K$ are the mean and Gaussian curvatures. (There are no higher-order terms, and in fact, this is already a consequence of Gauss-Bonnet.)

Now suppose you look at polyhedra $M$ instead of smooth surfaces. You can use the above power series in $\epsilon$ to define discrete notions of $H$ and $K$, even though the polyhedron is not even once-differentiable. The $K$ term will count the volume enclosed in red spherical caps around the vertices of the original polyhedron. You can derive the formula for $K$ in terms of the polyhedron angles, and you will see that the discrete Gauss curvature satisfies an exact analogue of Gauss-Bonnet, including counting the Euler characteristic of the polyhedron!
