nLab
Gauss-Bonnet theorem

Idea

The Chern-Gauss-Bonnet theorem gives a formula that computes the Euler characteristic of an even-dimensional smooth manifold as the integration of a curvature characteristic form of the Levi-Civita connection on its tangent bundle.

For surfaces the theorem simplifies and in this simpler version is the older Gauss-Bonnet theorem .

Statement

For surfaces

(…)

For smooth manifolds

Theorem

Let $X$ be a compact smooth manifold of even dimension $\dim X=2k$. Write $\chi (X)$ for its Euler characteristic.

For $\nabla$ any Levi-Civita connection on its tangent bundle, write $F_{\nabla }$ for its curvature 2-form, valued in the orthogonal Lie algebra $\mathrm{𝔰𝔬}(2k)$ and $\mathrm{Pf}(F_{\nabla })$ for its Pfaffian $2k$-form.

Then

$$\chi (X)={\left(\frac{-1}{2\pi }\right)}^k{\int }_X\mathrm{Pf}(F_{\nabla }).$$\chi(X) = \left( \frac{-1}{2 \pi} \right)^{k} \int_X Pf(F_\nabla) \,.

For orbifolds

There is a generalization for $X$ an orbifold due to (Satake).

Related concepts

• Euler characteristic

References

The Chern-Gauss-Bonnet theorem goes back to

• Shiing-shen Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds , Annals, 45 (1944), 747-752.

A classical textbook reference is chapter X of volume II of

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

Discussion is for instance in

• Denis Bell, The Gauss-Bonnet theorem for vector bundles (pdf)

Expositions include

• Liviu I. Nicolaescu, The many faces of the Gauss-Bonnet theorem (pdf)

• Denis Bell, The Gauss-Bonnet theorem (pdf)

• Chenchang Zhu, The Gauss-Bonnet theorem and its applications (pdf)

• Johannes Ebert, MO comment

The generalization to orbifolds is considered in

• Ichiro Satake, The Gauss-Bonnet Theorem for V-manifolds J. Math. Soc. Japan Volume 9, Number 4 (1957), 464-492. (ProjectEclid)