Chern-Gauss-Bonnet theorem

The Chern-Gauss-Bonnet theorem connects the shape's curvature to its overall topology, specifically its Euler characteristic, which is a number describing a surface's fundamental properties (like the number of holes). For a smooth, curved surface like a sphere or a torus, the theorem states that integrating certain curvature quantities over the entire surface results in a value directly related to this topological characteristic. In essence, it reveals a deep link between the geometric bending of a surface and its intrinsic shape properties, showing that the total curvature captures essential information about the surface's fundamental structure.