The Gauss-Bonnet theorem

The Gauss-Bonnet Theorem connects geometry and topology by relating the curvature of a surface to its shape. It states that for a curved surface, the total curvature (how much it bends) plus the total "genus" (number of holes) directly corresponds to a fixed value: the Euler characteristic. In simpler terms, it reveals that the way a surface curves is tied to its fundamental properties, like whether it has holes or not. This relationship helps mathematicians understand not just shapes but also the intrinsic nature of surfaces in different dimensions.