Milnor's theorem

Milnor's theorem relates the shape of a smooth, closed surface to a mathematical property called the total curvature. Specifically, it states that for such surfaces, the total curvature is directly linked to the surface’s topology—meaning its fundamental shape features, like the number of holes or handles it has. In essence, the theorem provides a precise way to connect the surface’s geometric bending (curvature) with its topological structure, helping mathematicians understand how the surface’s shape influences its geometric properties.