Gauss–Bonnet on surfaces

Let M be a compact connected oriented smooth Riemannian surface whose boundary is piecewise smooth and regular, with finitely many ordinary corners. Orient each boundary component positively, keeping the surface locally to the left. Along a smooth boundary arc, use unit tangent T and its positive quarter-turn N inside the tangent plane; signed geodesic curvature is $k_g=g(\nabla_TT,N)$. At a corner include the signed exterior turning angle from the incoming to outgoing tangent.

Gauss–Bonnet then gives the displayed sum. For a positively traversed convex corner of interior angle alpha, the exterior angle is π−alpha. Reentrant corners or reversed boundary motion change signs, so absolute curvature is not a substitute for the signed term. The metric area element is used in the interior integral. These conventions make a counterclockwise Euclidean circle bounding a disk have positive geodesic curvature, agreeing with the familiar full turn of a plane curve.

For a closed surface, meaning compact and without boundary here, only the interior integral remains. A sphere of radius ρ has K=1/ρ² and area 4πρ², giving total curvature 4π independently of radius. A closed oriented surface of genus h has Euler characteristic 2−2h, so its total curvature is 4π(1−h). The equality is global: it does not require curvature to have a fixed sign or to be constant.

A flat torus has zero curvature everywhere, consistent with Euler characteristic zero. A curved embedded torus can have both positive and negative regions whose integrals cancel, giving the same total. Conversely, a closed oriented surface with negative Euler characteristic cannot carry a metric with nonnegative Gaussian curvature everywhere, since its curvature integral must be negative. This obstruction follows directly from the sign of an integral; it does not classify all possible metrics or embeddings.

A flat Euclidean disk has K=0, yet Euler characteristic one. Its smooth circular boundary contributes 2π, supplying the entire right side. A planar polygon has straight boundary arcs with zero geodesic curvature, so its signed exterior angles supply that same total. A geodesic polygon on a curved surface instead places some contribution in the interior: for a topological disk with geodesic sides, the difference between angle sum and the Euclidean value measures its integrated Gaussian curvature.

The theorem can be motivated by subdividing a surface into small oriented pieces and expressing connection rotation as curvature. When pieces are combined, oppositely oriented interior edges cancel and the remaining combinatorial count becomes the Euler characteristic. This is a proof outline, not a full triangulation and connection-form argument. For computations, explicitly list which terms vanish and why; a geodesic boundary has zero geodesic curvature intrinsically even if it bends as a curve in the surrounding space.