Gauss-Bonnet Theorem

Prerequisites: Curvature, Topology. Full path (6 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Take a closed surface, like a sphere, a doughnut or a pretzel, and measure at every point how strongly it is curved. On a sphere the surface bends the same way in every direction, like the top of a ball. Around the inner ring of a doughnut it bends in opposite ways in two directions, like a saddle. Now add up this curvature over the whole surface.

The Gauss-Bonnet theorem says that the total depends only on the number of holes. If we dent the sphere, squash it into an egg, or stretch it into a long sausage, the curvature moves around: it piles up in some places and decreases in others, but the total stays exactly the same. Only cutting or gluing, which changes the number of holes, changes the total. For every surface that can be deformed into a sphere it is four pi; for every doughnut it is zero, because the positive curvature on the outside cancels the negative curvature on the inside.

A small-scale version is familiar from navigation on the Earth. A triangle drawn on a sphere with straight (great circle) sides has angles that add up to more than 180 degrees, and the excess is proportional to the area of the triangle times the curvature. Adding up such excesses over the whole surface gives the theorem.

The result is the simplest example of a pattern found all over physics and mathematics: an integral of a local quantity (curvature) gives a global integer that cannot change under smooth deformations.

Concrete

Let $M$ be a two-dimensional surface with a Riemannian metric, and let $K$ be its Gaussian curvature. For a surface in three-dimensional space, $K = \kappa_1 \kappa_2$ is the product of the two principal curvatures; Gauss showed that $K$ can be computed from the metric alone. For a sphere of radius $R$, $K = 1/R^2$; for a plane or a cylinder, $K = 0$; for a saddle, $K < 0$.

Local version. For a geodesic triangle $T$ with interior angles $\alpha,\beta,\gamma$, $$\alpha + \beta + \gamma - \pi = \int_T K\, dA .$$ On the unit sphere, the triangle with one vertex at the north pole and two vertices on the equator, $90^\circ$ apart, has three right angles. Its angle excess is $\pi/2$, and its area is one eighth of $4\pi$, which is also $\pi/2$. Equivalently, a vector parallel transported once around the boundary of $T$ comes back rotated by the angle $\int_T K\,dA$.

Global version. For a compact surface $M$ with boundary $\partial M$, $$\int_M K\, dA + \int_{\partial M} k_g\, ds = 2\pi\, \chi(M) ,$$ where $k_g$ is the geodesic curvature of the boundary curve and $\chi(M)$ is the Euler characteristic. For a closed surface (no boundary) the second term is absent. The Euler characteristic can be computed from any triangulation as $$\chi = V - E + F ,$$ with $V$, $E$, $F$ the numbers of vertices, edges and faces. For a closed orientable surface with $g$ handles, $\chi = 2 - 2g$.

Examples.

• Sphere of radius $R$: $\int K\,dA = \frac{1}{R^2}\cdot 4\pi R^2 = 4\pi = 2\pi\cdot 2$, matching $\chi(S^2) = 2$. A tetrahedron, a triangulation of the sphere, gives $V-E+F = 4 - 6 + 4 = 2$.
• Torus: $\chi = 0$, so $\int K\, dA = 0$ for every metric on the torus. In particular a torus cannot carry a metric with $K>0$ everywhere.
• Flat disk of radius $r$: $K = 0$ and the boundary circle has $k_g = 1/r$, so $\int_{\partial M} k_g\,ds = 2\pi$, matching $\chi(\text{disk}) = 1$.

In physics. In two dimensions the scalar curvature is $R = 2K$, so the Einstein-Hilbert action of a closed surface is a topological invariant: $$\frac{1}{4\pi}\int_M \sqrt{g}\, R\, d^2x = \chi(M) .$$ For this reason two-dimensional gravity has no local dynamics, and in string perturbation theory this term weights a worldsheet of genus $g$ by a power of the string coupling, $g_s^{-\chi} = g_s^{2g-2}$. The same structure, an integral of a curvature that gives an integer, appears for the Berry curvature of a band in the Brillouin zone, where it gives the Chern number of the quantum Hall effect.

• A fantastic introduction that explains the Gauss-Bonnet theorem in intuitive terms is Geometry and topology in many-particle systems by Joel E. Moore

Abstract

The theorem generalises to any closed oriented Riemannian manifold of even dimension $2n$ as the Chern-Gauss-Bonnet theorem. Let $\Omega^{a}{}_{b}$ be the curvature 2-form of the Levi-Civita connection in an oriented orthonormal frame, an $\mathfrak{so}(2n)$-valued 2-form. Its Pfaffian is $$\operatorname{Pf}(\Omega) = \frac{1}{2^n\, n!}\, \epsilon_{a_1 a_2 \dots a_{2n-1} a_{2n}}\, \Omega^{a_1 a_2}\wedge\cdots\wedge\Omega^{a_{2n-1}a_{2n}} ,$$ a $2n$-form that does not depend on the choice of frame. The theorem states $$\chi(M) = \frac{1}{(2\pi)^n}\int_M \operatorname{Pf}(\Omega) .$$ For $n = 1$, $\operatorname{Pf}(\Omega) = \Omega^{12} = K\, dA$ (with orientation conventions such that the sphere has positive curvature), which is the classical theorem. In four dimensions it reads $$\chi(M) = \frac{1}{32\pi^2}\int_M d^4x\,\sqrt{g}\,\left( R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} - 4R_{\mu\nu}R^{\mu\nu} + R^2 \right) ,$$ which is why the Gauss-Bonnet term added to a four-dimensional gravitational action does not change the classical equations of motion. In higher dimensions the same combination is the second Lovelock term.

Structure. In Chern-Weil theory, the closed form $\operatorname{Pf}(\Omega/2\pi)$ represents the Euler class $e(TM) \in H^{2n}(M;\mathbb{R})$ of the tangent bundle, and a different metric changes the form by an exact form. The topological side is the Poincaré-Hopf theorem: for a vector field with isolated zeros, the sum of the indices of the zeros equals $\chi(M)$, and $\int_M e(TM)$ equals the self-intersection number of the zero section of $TM$. Chern's intrinsic proof transgresses $\operatorname{Pf}(\Omega)$ on the unit sphere bundle, where it becomes exact, and reduces the integral to a sum over the zeros of a vector field.

The theorem is the special case of the Atiyah-Singer index theorem for the operator $d + d^\dagger$ from even to odd forms, whose index is $\sum_k (-1)^k b_k = \chi(M)$ by Hodge theory. For manifolds with boundary, a boundary term built from the second fundamental form generalises the geodesic curvature term.

Why is it interesting?

The Gauss-Bonnet theorem is a formula that yields a topological invariant, i.e. something that can be used to characterise e.g. manifolds.

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