The Gauss-Bonnet Theorem

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 ($K = 0$): $\int K\, dA = 0$. ✓
• Genus 2 surface: $\chi = -2$, so total curvature $= -4\pi$.

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.

Example 8.1 (Geodesic Triangle on a Sphere). Consider a geodesic triangle on $S^2$ with interior angles $\theta_1, \theta_2, \theta_3$. The area is $A = \theta_1 + \theta_2 + \theta_3 - \pi$. This is a special case: $\int_M K\, dA = 1 \cdot A$ (since $K = 1$), the geodesic curvature term vanishes, and the exterior angles are $\pi - \theta_i$, so:

$A + \sum(\pi - \theta_i) = A + 3\pi - (\theta_1 + \theta_2 + \theta_3) = 2\pi \cdot 1 = 2\pi$

This confirms $A = \theta_1 + \theta_2 + \theta_3 - \pi$.

Step 1: Triangulation. Triangulate $M$ into geodesic triangles. Let $V, E, F$ be the numbers of vertices, edges, and faces, with $\chi(M) = V - E + F$.

Step 2: Apply Gauss-Bonnet to each triangle. For each geodesic triangle $T$ with interior angles $\alpha, \beta, \gamma$:

$\int_T K\, dA = (\alpha + \beta + \gamma) - \pi$

This follows from the local Gauss-Bonnet formula for a geodesic triangle.

Step 3: Sum over all triangles. Summing over $F$ triangles:

$\int_M K\, dA = \sum_{i=1}^F (\alpha_i + \beta_i + \gamma_i) - F\pi$

Step 4: Relate angle sum to Euler characteristic. Each interior angle at a vertex appears once per incident triangle. The sum of all angles around a vertex is $2\pi$, so:

$\sum_{i=1}^F (\alpha_i + \beta_i + \gamma_i) = 2\pi V$

Thus $\int_M K\, dA = 2\pi V - F\pi$. Using $3F = 2E$ (each edge shared by 2 triangles) and $\chi = V - E + F$, we get $\int_M K\, dA = 2\pi \chi(M)$. $\blacksquare$

Corollary 8.4 (Curvature Sign and Topology).

• If $K > 0$ everywhere, then $\chi(M) > 0$, so $M$ is homeomorphic to $S^2$.
• If $K = 0$ everywhere, then $\chi(M) = 0$, so $M$ is homeomorphic to a torus $T^2$.
• If $K < 0$ everywhere, then $\chi(M) < 0$, so $M$ has genus $g \geq 2$.

Corollary 8.5 (Uniformization). Every compact Riemann surface admits a metric of constant curvature $K = 1, 0,$ or $-1$, depending on its genus. This is the uniformization theorem for Riemann surfaces.

Theorem 8.5 (Chern 1944). Let $M$ be a compact oriented Riemannian $2n$-manifold. Then:

$\int_M \mathrm{Pf}\left(\frac{\Omega}{2\pi}\right) = \chi(M)$

where $\Omega$ is the curvature 2-form of the Levi-Civita connection and $\mathrm{Pf}$ is the Pfaffian. For surfaces ($n = 1$), $\mathrm{Pf}(\Omega/2\pi) = K\, dA/2\pi$, recovering the classical theorem.

In terms of the Riemann curvature tensor $R_{ijkl}$:

$\chi(M) = \frac{1}{2^{2n}\pi^{n}n!} \int_M \epsilon^{i_1\ldots i_{2n}} \Omega_{i_1i_2} \wedge \cdots \wedge \Omega_{i_{2n-1}i_{2n}}$

Example 8.2. For $M = S^4$ (4-sphere) with the round metric, $\chi(S^4) = 2$. The integrand is a 4-form constructed from the curvature, and $\int_{S^4} \mathrm{Pf}(\Omega/2\pi) = 2$.

Application 1: Topological obstructions to metrics. A manifold that admits a metric with $K > 0$ must have $\chi(M) > 0$ for surfaces. In higher dimensions, obstructions involve the A-genus and Dirac operators (Lichnerowicz theorem, Hitchin’s work).

Application 2: The Gauss-Bonnet theorem as an index theorem. The Gauss-Bonnet theorem is a special case of the Atiyah-Singer index theorem for the de Rham complex. The Euler characteristic is the index of $d + d^*$ acting on differential forms.

Application 3: Geometric inequalities. For a compact surface $M$ with area $A$ and Gaussian curvature bounded by $|K| \leq C$:

$|\chi(M)| \leq \frac{C A}{2\pi}$

This follows directly from $2\pi|\chi| = |\int K| \leq C A$.

Problem 1. Prove that there is no metric of strictly positive Gaussian curvature on a torus.

Solution. The Gauss-Bonnet theorem gives $\int_{T^2} K\, dA = 2\pi \chi(T^2) = 0$. If $K > 0$ everywhere, the integral would be strictly positive. Contradiction. $\blacksquare$

Problem 2. A geodesic hexagon on a surface has six geodesic edges meeting at right angles. If the surface has constant curvature $K = -1$, find the area of the hexagon.

Solution. For a geodesic polygon with $n$ sides and interior angles $\theta_i$ on a surface with constant curvature $K = -1$: $\int_T K\, dA = \sum \theta_i - (n-2)\pi$. For right angles: $\theta_i = \pi/2$, $n = 6$, so $-\mathrm{Area} = 6(\pi/2) - 4\pi = 3\pi - 4\pi = -\pi$, giving $\mathrm{Area} = \pi$. $\blacksquare$

1. Compute the Euler characteristic of a compact surface of genus 3. What is its total curvature?
2. Show that a metric on $S^2$ with $K \geq 1$ has area $\leq 4\pi$.
3. Prove that any metric on $S^2$ has at least one point with $K > 0$.
4. Use Gauss-Bonnet with boundary to find the area of a spherical lune (region between two great circles) with angle $\theta$.
5. Show that $\chi(M)$ is even for every compact oriented 2-manifold.