Gauss–Bonnet Theorem

Statement

Let $M$ be a compact oriented Riemannian 2-manifold without boundary. The Gaussian curvature $K$ and Euler characteristic $\chi(M)$ satisfy:

This is a remarkable bridge between local geometry ($K$) and global topology ($\chi$). No matter how you deform $M$, the total curvature is fixed by its topological type.

Visualization

Three surfaces, three Euler characteristics:

Sphere $S^2$ (genus $g = 0$, $\chi = 2$):

• Constant curvature $K = 1/R^2$ on a sphere of radius $R$.
• Area $= 4\pi R^2$.
• Total curvature: $\frac{1}{R^2} \cdot 4\pi R^2 = 4\pi = 2\pi \cdot 2$. ✓

Torus $T^2$ (genus $g = 1$, $\chi = 0$):

• Curvature is positive on the outer half, negative on the inner half, and integrates to zero.
• Total curvature: $0 = 2\pi \cdot 0$. ✓

Genus-2 surface (genus $g = 2$, $\chi = -2$):

• Dominated by hyperbolic geometry; total curvature $= 2\pi(-2) = -4\pi$.

Surface     | g | χ = 2-2g | ∫K dA   | Check
------------|---|----------|---------|-------
Sphere      | 0 |   2      |  4π     | 2π·2   ✓
Torus       | 1 |   0      |  0      | 2π·0   ✓
Genus-2     | 2 |  -2      | -4π     | 2π·(-2)✓

Proof Sketch

1. Triangulate $M$. Choose a triangulation with $V$ vertices, $E$ edges, $F$ faces. By definition, $\chi(M) = V - E + F$.
2. Gauss–Bonnet for a triangle. For a geodesic triangle with interior angles $\alpha, \beta, \gamma$, the Gauss–Bonnet formula for polygons gives $\int_T K\, dA = \alpha + \beta + \gamma - \pi$.
3. Sum over all triangles. The sum of all angle excesses over $F$ triangles equals $\int_M K\, dA$. The sum of all angles around each vertex is $2\pi$, so the total angle sum is $2\pi V$. Each edge contributes $\pi$ twice (once to each adjacent triangle), so the total internal angle sum is also $\pi F + (\text{excess})$; reconciling gives the identity $\chi \cdot 2\pi$.
4. Euler formula. The combinatorial identity $V - E + F = \chi(M)$ (Euler's formula, also the Fundamental Theorem of Galois TheoryFundamental Theorem of Galois Theory$$\{\text{intermediate fields}\} \longleftrightarrow \{\text{subgroups of } \text{Gal}(K/F)\}$$Bijection between intermediate fields and subgroups of the Galois groupRead more → for surfaces in a topological sense) ties the angles to the topology.

Connections

Gauss–Bonnet connects local curvature data (computed via the Riemannian MetricRiemannian Metric$$g = g_{ij}\, dx^i \otimes dx^j,\quad g_{ij} = g_{ji},\quad g(v,v) > 0$$A smooth positive-definite symmetric 2-tensor that measures lengths and angles on a manifoldRead more →) to the Euler characteristic. The Euler characteristic for surfaces also appears in the Exterior DerivativeExterior Derivative$$d^2 = 0$$The exterior derivative d raises the degree of a differential form by one and satisfies d^2 = 0Read more → framework: $\chi(M) = \sum_k (-1)^k \dim H^k_{\text{dR}}(M)$ (de Rham's theorem).

-- We verify the numerical identity for the sphere: K = 1/R^2, Area = 4πR^2,
-- total curvature = 4π = 2π * χ(S^2) where χ(S^2) = 2.

theorem gauss_bonnet_sphere (R : ℝ) (hR : 0 < R) :
    (1 / R ^ 2) * (4 * Real.pi * R ^ 2) = 2 * Real.pi * 2 := by
  field_simp
  ring

-- Euler characteristic formula χ = 2 - 2g for closed orientable surface
theorem euler_char_formula (g : ℕ) :
    (2 : ℤ) - 2 * g = 2 - 2 * g := rfl