GaussBonnetChern

The Gauss-Bonnet-Chern theorem is a fundamental result in differential geometry that relates the integral of a curvature scalar over a compact manifold to its Euler characteristic. In its simplest form for a 2-dimensional surface without boundary, the theorem states that the integral of the Gaussian curvature K over the surface S is equal to 2 times its Euler characteristic chi(S). Mathematically, this is expressed as the integral of K dA = 2 pi chi(S).

The Euler characteristic is a topological invariant that can be calculated by triangulating the surface and counting vertices, edges, and faces. For a sphere, the Euler characteristic is 2, while for a torus, it is 0. This means that the total Gaussian curvature of a sphere must always be 4 pi, regardless of its specific shape. For a torus, the total Gaussian curvature is zero, implying that regions of positive and negative curvature must balance out.

The theorem can be generalized to higher dimensions and also incorporates boundary terms. The Chern-Gauss-Bonnet theorem is the most general form, relating the integral of a specific differential form (the Euler form) over a compact manifold with boundary to the Euler characteristic of the manifold and an integral over its boundary. This theorem has profound implications in various fields, including general relativity, where it relates the total mass-energy of an isolated system to the topology of spacetime, and quantum field theory.