The Gauss-Bonnet Theorem

8.1 Statement for Surfaces

Theorem 8.1 (Gauss-Bonnet, Global). Let $(M, g)$ be a compact, oriented Riemannian 2-manifold without boundary. Then:

$\int_M K\, dA = 2\pi \chi(M)$

where $K$ is the Gaussian curvature, $dA$ is the area form, and $\chi(M)$ is the Euler characteristic.

Corollary 8.2. The total curvature of a compact surface depends only on its topology, not on the metric.

Examples.

• $S^2$: $\int_{S^2} K\, dA = 2\pi \cdot 2 = 4\pi$. For the standard metric ($K = 1$): $\int K\, dA = 1 \cdot 4\pi = 4\pi$. ✓
• $T^2$ (torus): $\int_{T^2} K\, dA = 2\pi \cdot 0 = 0$. For the flat metric ($K = 0$): $\int K\, dA = 0$. ✓
• $S^1 \times S^1$ (genus 2): $\chi = -2$, so total curvature $= -4\pi$.

8.2 Gauss-Bonnet with Boundary

Theorem 8.3. Let $M$ be a compact oriented Riemannian 2-manifold with boundary $\partial M$ consisting of smooth curves meeting at exterior angles $\alpha_1, \ldots, \alpha_k$. Then:

$\int_M K\, dA + \int_{\partial M} \kappa_g\, ds + \sum_{i=1}^k \alpha_i = 2\pi \chi(M)$

where $\kappa_g$ is the geodesic curvature of the boundary.