The Gauss–Bonnet theorem

The Gauss–Bonnet theorem is a crowning result of surface theory that gives a fundamental connection between geometry and topology. Roughly speaking, geometry refers to the “local” properties—lengths, angles, curvature— of some fixed object, while topology seeks to identify the “global” properties that are unchanged by a continuous deformation, such as stretching or twisting. The theorem formalizes an intuitive idea: continuous changes to curvature on one region of a surface will be balanced out elsewhere, so the total curvature of the surface stays the same. Explicitly, the Gauss–Bonnet theorem says that a surface’s total curvature, defined using its local Gaussian curvature, is directly proportional to the number of holes in the surface, which comes from an invariant quantity called its Euler characteristic. The Euler characteristic is a way of classifying which surfaces can be continuously deformed into one another; as an informal example, the classic joke that “a topologist is a person who cannot tell the difference between a coffee mug and a doughnut” comes from the fact that the objects each have one hole. Even though a coffee mug and a doughnut have visibly different geometric shapes, according to the Gauss–Bonnet theorem, both objects will have the same total curvature. The proof itself is delightfully systematic: we first find the total curvature of a curve on a plane, extend that result to curves on three-dimensional surfaces, extend that result to “polygons” on surfaces, and finally the entire surface. In Section 2, we prove Hopf’s Umlaufsatz for the total curvature of a simple closed curve in R2. Sections 3, 4, and 5 introduce concepts from differential geometry to define Gaussian curvature. In Section 6, we prove the local Gauss–Bonnet theorem for the total curvature of a surface polygon. At last, in Section 7, we prove the global Gauss–Bonnet theorem for compact surfaces by covering the surface with polygons and applying the local Gauss–Bonnet theorem to each one. Our discussion focuses on exposition, and references will be given in place of tedious computations when reasonable. This paper assumes a somewhat rigorous understanding of multivariable calculus and linear algebra, as well as some elementary group theory.