Why should I care about Gauss-Bonnet (and Gaussian curvature)? The Gauss-Bonnet Theorem (for orient

In retrospect, this post got quite long. Also, the level varies greatly - sorry! Feel free to ask any questions. I guess I am quite fond of this topic, even though I am not as knowledgeable on it as other people on here. Anyway, hopefully this is helpful to someone :-)
In my opinion this theorem is one of the crown jewels of mathematics.
There are a number of ways to interpret or generalize this statement. For instance, since $2\mathrm{\pi <!--\; \pi \; -->}\mathrm{\chi <!--\; \chi \; -->}(M)$ is a purely topological quantity, this tells us that stretching or deforming M through smooth isotopies will not change the integral of the Gaussian curvature. So, we could take a sphere $S^2(r)$of radius r in ${\mathbb{R}}^3$. Then the Gaussian curvature is everywhere $\frac{1}{r^2}$. The integral is
${\int <!--\; \int \; -->}_{S^2(r)}\frac{1}{r^2}\mathrm{v}\mathrm{o}\mathrm{l}=\frac{4\mathrm{\pi <!--\; \pi \; -->}{r}^2}{r^2}=4\mathrm{\pi <!--\; \pi \; -->}=2\mathrm{\pi <!--\; \pi \; -->}\mathrm{\chi <!--\; \chi \; -->}(S^2(r)).$
Now, we can deform this sphere $S^2(r)$ as we wish through isotopies and we see that the Gauss-Bonnet theorem gives us a "conservation law":
if we deform our surface M in a volume-preserving manner such a way that the curvature in one region becomes greater, then the curvature in another region must become smaller.
This is a nice intuition for at least some of what the theorem is saying. However, there is another interpretation. Indeed, we have a classical topological invariant χ(M), and with some work we were able to find a "geometric quantity" to integrate to return χ(M), establishing a connection between a global geometric quantity and a topological quantity. This can be interpreted as
${\int <!--\; \int \; -->}_M\frac{K}{2\mathrm{\pi <!--\; \pi \; -->}}\mathrm{v}\mathrm{o}\mathrm{l}=\mathrm{\chi <!--\; \chi \; -->}(M).$.
Now, the natural objects of integration on manifolds are differential forms and indeed we have integrated a 2−form $\frac{K}{2\mathrm{\pi <!--\; \pi \; -->}}\mathrm{v}\mathrm{o}\mathrm{l}$ vol to return χ(M). This is a closed 2−form and defines a cohomology class $\mathrm{\omega <!--\; \omega \; -->}\in <!--\; \in \; -->H^2(M,\mathbb{R})$.
So, we arrive at a natural question:
Given a compact n−manifold M, can we find a cohomology class
$\mathrm{\omega <!--\; \omega \; -->}\in <!--\; \in \; -->H^n(M,\mathbb{R})$ so that ${\int <!--\; \int \; -->}_M\mathrm{\omega <!--\; \omega \; -->}=\mathrm{\chi <!--\; \chi \; -->}(M)$?
For n=2, the result is Gauss-Bonnet. If n is odd, it is an easy consequence of Poincaré Duality that χ(M)=0, so the answer is yes but for stupid reasons: simply take $0\in <!--\; \in \; -->H^n(M,\mathbb{R})$. For n even, the question is more interesting. We next note that we had more structure in the case of the Gauss-Bonnet theorem. Indeed, we were using the Riemannian structure coming from our manifold being in ${\mathbb{R}}^3$. So, we can equip M with a Riemannian structure (M,g). Now that we have done that, we would like to introduce a notion of curvature. Curvature is - in some sense - a second order phenomenon, so we need to introduce a notion of a second derivative. The method of doing this is to introduce an affine connection on TM. This is an operator
$\mathrm{\nabla <!--\; \nabla \; -->}:\mathfrak{X}(M)\times <!--\; \times \; -->\mathfrak{X}(M)\to <!--\; \to \; -->\mathfrak{X}(M)$
written as $\mathrm{\nabla <!--\; \nabla \; -->}(X,Y)={\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Y$ This has some natural properties:
1. $\mathrm{\nabla <!--\; \nabla \; -->}$ is $\mathbb{R}-<!--\; -\; -->$linear in both X and Y
2. For any ${\mathrm{\nabla <!--\; \nabla \; -->}}_{fX}Y=f{\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Y$
3. For any ${\mathrm{\nabla <!--\; \nabla \; -->}}_X(fY)=X(f)Y+f{\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Y$ which we call the Leibniz property.
The standard example of this is the directional derivative of a vector field $Y\in <!--\; \in \; -->\mathfrak{X}({\mathbb{R}}^3)$ with respect to another vector field X, written in multivariable calculus by $D_{X}Y$. Anyway, associated to this $\mathrm{\nabla <!--\; \nabla \; -->}$ are a pair of tensors:
$T(X,Y)={\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Y-<!--\; -\; -->{\mathrm{\nabla <!--\; \nabla \; -->}}_{Y}X-<!--\; -\; -->[X,Y]$
called the torsion and
$R(X,Y)Z={\mathrm{\nabla <!--\; \nabla \; -->}}_X{\mathrm{\nabla <!--\; \nabla \; -->}}_{Y}Z-<!--\; -\; -->{\mathrm{\nabla <!--\; \nabla \; -->}}_Y{\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Z-<!--\; -\; -->{\mathrm{\nabla <!--\; \nabla \; -->}}_{[X,Y]}Z$
called the curvature. There is a theorem which says that on a Riemannian manifold (M,g), there is a unique connection $\mathrm{\nabla <!--\; \nabla \; -->}$ satisfying ${\mathrm{\nabla <!--\; \nabla \; -->}}_{X}g(Y,Z)=g({\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Y,Z)+g(Y,{\mathrm{\nabla <!--\; \nabla \; -->}}_{X}Z)$ This connection is called the Levi-Civita connection. The virtue of this is that there is a canonical connection associated to a Riemannian manifold, which will turn out to be quite important.
With this in hand, we can take our Riemannian manifold (M,g) with its LC connection $\mathrm{\nabla <!--\; \nabla \; -->}$ and study the properties of its curvature. (By the way, you can recover the Gaussian curvature from this mysterious R in the case of a surface, but I won't explain how.) If we take a trivialization of the tangent bundle by a local frame $(e_1,\dots <!--\; \dots \; -->,e_n)$, we can describe the data of the connection using the equations:
${\mathrm{\nabla <!--\; \nabla \; -->}}_X{e}_j=\underset{i}{\sum <!--\; \sum \; -->}{\mathrm{\omega <!--\; \omega \; -->}}_j^i(X)e_i$
where ${\mathrm{\omega <!--\; \omega \; -->}}_j^i$ are differential 1−forms. Furthermore, we can describe the data of the curvature using
$1\le <!--\; \le \; -->i\le <!--\; \le \; -->n$
for all $1\le <!--\; \le \; -->i\le <!--\; \le \; -->n$, where the ${\mathrm{\Omega <!--\; \Omega \; -->}}_j^i$ are 2−forms. In particular, we get matrices $\mathrm{\omega <!--\; \omega \; -->}=({\mathrm{\omega <!--\; \omega \; -->}}_j^i)$ and $\mathrm{\Omega <!--\; \Omega \; -->}=({\mathrm{\Omega <!--\; \Omega \; -->}}_j^i)$ describing the connection and the curvature, respectively, in a local trivialization. We would like to assemble from these guys a differential form that we can integrate to get back χ(M). The bridge is provided by the Chern-Weil homomorphism, which allows us to construct cohomology classes from these matrices of differential forms. The statement is
Theorem (Chern-Weil) Suppose E is a rank r vector bundle with connection $\mathrm{\nabla <!--\; \nabla \; -->}$. Suppose P is a $\mathrm{G}\mathrm{L}(r,\mathbb{R})-<!--\; -\; -->$−invariant polynomial on $\mathfrak{g}\mathfrak{l}(r,\mathbb{R})$ of degree k. Then, the 2k−form $P(\mathrm{\Omega <!--\; \Omega \; -->})$ on M is globally defined, closed, and $[P(\mathrm{\Omega <!--\; \Omega \; -->})]\in <!--\; \in \; -->H^{2k}(M,\mathbb{R})$ is independent of the connection.
The definition of a connection on E is analogous to the above. A polynomial on $\mathfrak{g}\mathfrak{l}(r,\mathbb{R})$ means a polynomial that takes an$r\times <!--\; \times \; -->r$ matrix $X=(x_j^i)$ as its input. Anyway, this theorem tells us that we can construct cohomology classes explicitly from the information of the curvature of a connection on a vector bundle on M. Moreover, the cohomology class is independent of the choice of connection. One way to interpret this is that we are getting a way to construct distinguished representatives of our cohomology classes on M. This is a motif that appears frequently in geometry. This lets us state our final theorem.
(Chern-Gauss-Bonnet) Let (M,g) denote a compact Riemannian manifold of dimension 2n. Then
${\int <!--\; \int \; -->}_M\frac{1}{(2\mathrm{\pi <!--\; \pi \; -->}{)}^n}\mathrm{P}\mathrm{f}(\mathrm{\Omega <!--\; \Omega \; -->})=\mathrm{\chi <!--\; \chi \; -->}(M).$
Here, Pf denotes the Pfaffian, which is a certain polynomial defined on so$\mathfrak{s}\mathfrak{o}(2n,\mathbb{R})$ characterized (up to a sign) by ${\mathrm{P}\mathrm{f}}^2=det$=det as polynomial functions. I have swept a lot of details under the rug here - but the moral of the story is that the presence of a Riemannian metric gives a version of the above Chern-Weil theorem for SO(2n)−invariant polynomials on $\mathfrak{s}\mathfrak{o}(2n,\mathbb{R})$.
Lastly, here is a high level explanation of what we are doing here. In topology, one associates to a vector bundle $E\to <!--\; \to \; -->M$ characteristic classes, which are cohomology classes in H∗(M) that are invariants of the bundle. I won't get too much into this, but there is a characteristic class of oriented real vector bundles called the Euler class, e(E). It is suggestively named because in the case where $e(E)\in <!--\; \in \; -->H^n(M,\mathbb{R})$(viewed for instance as a de Rham cohomology class) satisfies
${\int <!--\; \int \; -->}_{M}e(E)=\mathrm{\chi <!--\; \chi \; -->}(M).$
So, we have really constructed an explicit form $\mathrm{P}\mathrm{f}(\mathrm{\Omega <!--\; \Omega \; -->})$ on M representing this cohomology class e(E), and that is the content of the theorem. It turns out that characteristic classes of oriented bundles rank r can be defined as cohomology classes of the classifying space BSO(r). It can be proven that the cohomology of BG (for G a compact Lie group) is $H^{\ast <!--\; \ast \; -->}(BG,\mathbb{R})\cong <!--\; \cong \; -->\mathbb{R}[\mathfrak{g}{]}^G$, i.e. the invariant polynomials on the Lie algebra g. So, the Chern-Weil homomorphism in its most general form expresses characteristic classes of a (say) oriented real bundle of rank r in terms of polynomials of the curvature. So, in this setup the Gauss-Bonnet theorem is a special case of the fact that the polynomial Pf computes the Euler class.
If this is of interest to you, I started learning about this in Tu's Differential Geometry: Connections, Curvature, and Characteristic classes, which is a clear and pleasant read for someone with a basic knowledge of manifolds and de Rham cohomology.