Gauss–Bonnet theorem

In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Suppose M is a compact two-dimensional Riemannian manifold with boundary ∂M. Let K be the Gaussian curvature of M, and let kg be the geodesic curvature of ∂M. Then where dA is the element of area of the surface, and ds is the line element along the boundary of M. Here, χ(M) is the Euler characteristic of M. If the boundary ∂M is piecewise smooth, then we interpret the integral ∫∂M kg ds as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary. Many standard proofs use the theorem of turning tangents, which states roughly that the winding number of a Jordan curve is exactly ±1. Suppose M is the northern hemisphere cut out from a sphere of radius R. Its Euler characteristic is 1. On the left hand side of the theorem, we have and , because the boundary is the equator and the equator is a geodesic of the sphere. Then . On the other hand, suppose we flatten the hemisphere to make it into a disk. This transformation is a homeomorphism, so the Euler characteristic is still 1. However, on the left hand side of the theorem we now have and , because a circumference is not a geodesic of the plane. Then . Finally, take a sphere octant, also homeomorphic to the previous cases. Then . Now almost everywhere along the border, which is a geodesic triangle. But we have three right-angle corners, so .