Theorem of Gauss-Bonnet

The Gauss-Bonnet theorem links geometry and topology by relating the total curvature of a surface to its shape. Specifically, for a smooth, curved surface with edges, the theorem states that the sum of the surface's angular deficits (or excesses), plus the total curvature integrated over the surface, equals 2π times the surface's overall shape characteristic called its Euler characteristic. This reveals that the way a surface bends and its fundamental shape are deeply connected: the total "bendiness" reflects how many holes or handles the surface has, uniting geometry (curvature) with topology (shape features).