system
Given the following post, respond as if you're commenting on Reddit:
human
Is there an intuitive way to understand the Gauss-Bonnet theorem?
gpt
Gauss-Bonnet identifies a fundamental connection between geometry and topology. And, perhaps more crucially, between local properties and global properties.

In a sort of naive sense, topology is about ignoring geometry (topology is much more subtle and complicated than this, but roll with me here). Given a topology, what about a space/surface/what-have-you stays the same given continuous transformations, i.e. stretching and pulling and twisting? The classic example is recognizing that a coffee cup and a donut are the same topological "shape".

Geometry is generally much more concrete. You're measuring angles and curves and distances, the properties of an immutable object.

What the Gauss-Bonnet Theorem is saying is that the total curvature--defined as the integral of the Gaussian curvature--of a surface is directly proportional to its Euler characteristic, which is basically the number of "holes" it has in it. The first is a geometric property defined locally. The second is a topological property defined globally. This is a kinda ridiculous connection because you can do *crazy* things to a surface with continuous transformations. Completely change what it looks like. But because the Euler characteristic is constant, the Gauss Bonnet theorem says that the total curvature won't change.

Intuitively this really just means that, no matter how you do it, continuous changes to the curvature of a surface will be balanced out elsewhere on the surface, so that the total curvature remains unchanged.