 Gauss–Bonnet theorem

For a compact oriented Riemannian surface $D$ with piecewise smooth boundary,Here $K$ is Gaussian curvature, $dA$ the area element of a surface, $k_g$ signed geodesic curvature, and $\chi (D)$ the Euler characteristic. Traverse each boundary component with the surface on the left. The signed exterior angle at a convex corner of interior angle ${\alpha }_j$ is ${\epsilon }_j=\pi -{\alpha }_j$. Empty boundaries contribute no boundary or corner terms.^[ref]

On a disk in the Euclidean plane, zero interior Gaussian curvature leaves total boundary turning $2\pi$. On the unit sphere, $K=1$ and $\chi =2$, giving area $4\pi$. A geodesic triangle has no smooth boundary term, giving the spherical excess formula or hyperbolic angle defect.

For a closed surface, a local metric quantity integrates to a topological invariant. A proof triangulates the surface and applies the local turning formula to each triangle. Shared-edge contributions cancel and the angle sums leave $2\pi (V-E+F)$. Thus changing the Riemannian metric cannot change total Gaussian curvature.^[ref]

The Gauss–Bonnet theorem is a fundamental result in differential geometry that relates the geometry of a surface to its topology. It provides a connection between the curvature of a surface and its Euler characteristic, which is a topological invariant.