nLab
Gauss-Bonnet theorem

Context

\infty-Chern-Weil theory

Differential cohomology

Index theory

Contents

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 XX be a compact smooth manifold of even dimension dimX=2kdim X = 2k. Write χ(X)\chi(X) for its Euler characteristic.

For \nabla any Levi-Civita connection on its tangent bundle, write F F_\nabla for its curvature 2-form, valued in the orthogonal Lie algebra 𝔰𝔬(2k)\mathfrak{so}(2k) and Pf(F )Pf(F_\nabla) for its Pfaffian 2k2k-form.

Then

χ(X)=(12π) k XPf(F ). \chi(X) = \left( \frac{-1}{2 \pi} \right)^{k} \int_X Pf(F_\nabla) \,.

For orbifolds

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

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

Discussion is for instance in

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

Expositions include

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)

Revised on May 20, 2013 12:46:30 by Urs Schreiber (89.204.130.66)