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 $\mathfrak{so}(2k)$ and $Pf(F_\nabla)$ for its Pfaffian $2k$-form.

Then

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

For orbifolds

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

Related concepts

• Euler characteristic

• Euler form

• Poincaré–Hopf theorem

References

Named after Carl Friedrich Gauss.

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 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)

Generalization to orbifolds:

• Christopher Seaton, Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary, Differential Geometry and its Applications Volume 26, Issue 1, February 2008, Pages 42-51 (doi:10.1016/j.difgeo.2007.11.002)

For an approach via 0|2-dimensional supersymmetric Euclidean field theory?, see

• Daniel Berwick-Evans, The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories, Communications in Mathematical Physics, Volume 335 (2015), (arXiv:1310.5383).