TITLE: Why should I care about Gauss-Bonnet (and Gaussian curvature)?
QUESTION [25 upvotes]: The Gauss-Bonnet Theorem (for orientable surfaces without boundary) states that for surface $M$, with Gaussian curvature at a point $K$, we have
$$\int_M K\ dA=2\pi\chi(M).$$
Right now, this just says to me that the integral of something I don’t care about is equal to the ratio of a circumference to radius of a (Euclidean) circle times something I do care about. I get why normal curvature is interesting. I get Gaussian curvature is important in the Theorema Egregium and classification of surfaces of constant curvature. In the former, it helps classify surfaces up to isometry and show that developable surfaces are ruled. In the latter, it tells us that Riemannian manifolds with transitive isometry groups locally have a structure from a very small selection. But that’s just Gaussian curvature giving us other things. I don’t understand why I should like it for its own sake, and as such, why I should care about Gauss-Bonnet. I get that it links differential geometry and topology, but maybe there are other ways to do this, and this one feels a little convoluted, and topology is already linked to differential manifolds, and thus Riemannian manifolds, through Poincare-Hopf.

REPLY [3 votes]: I think one does not need to care about the notion of curvature or even Euler characteristics to find Gauss-Bonnet at least surprising.
A manifold is just a topological space. But when you make it into a Riemannan manifold, you add some additional structure. And it is reasonable to ask "in how many different ways can I do this?". Visually, a Riemannian structure on a, say, topological sphere makes it into some more or less deformed potatoe shape. And this can get pretty wild up to a point where it does not embed into $\Bbb R^3$ anymore. So are there any limits at all?
Note the following two things about Gauss-Bonnet.

Whatever curvature is, it is something that is defined purely in terms of the Riemannian structure, and it is completely determined by it.
When you integrate it over the manifold, it does not depend on the Riemannian structure.

Isn't that surprising? We just uncovered that Riemannian structures cannot be chosen completely arbitrary.