Knowlpedia

Theorem

Chern–Gauss–Bonnet theorem

The integral of the Euler curvature form of a closed oriented even-dimensional Riemannian manifold equals its Euler characteristic.

Statement

Let MM be a closed oriented of dimension 2m2m, and let Ω\Omega be the curvature matrix of its in a local oriented orthonormal frame. With the Chern–Weil normalization

e(TM,∇LC)=Pf⁡ ⁣(Ω2π),e(TM,\nabla^{\mathrm{LC}})=\operatorname{Pf}\!\left(\frac{\Omega}{2\pi}\right),

the Chern–Gauss–Bonnet theorem states

∫Me(TM,∇LC)=⟨e(TM),[M]⟩=χ(M).\int_M e(TM,\nabla^{\mathrm{LC}}) =\left\langle e(TM),[M]\right\rangle =\chi(M).

Thus the integral of this metric-dependent curvature form is the topological Euler characteristic of MM. The middle expression uses the topological Euler class and the chosen orientation.

Why the equality is topological

The Pfaffian is an invariant polynomial on the even-dimensional . Chern–Weil theory therefore makes the displayed Euler form closed and identifies its de Rham class with the of TMTM. Evaluation on the is independent of the metric and connection. The remaining equality with χ(M)\chi(M) is the topological characterization of the Euler number.

Chern’s intrinsic proof transgresses the Euler form to the unit tangent and applies Stokes’s theorem around the isolated zeros of a ; their local indices sum to the Euler characteristic Chern, §§1–3.

Surface case

For an oriented closed Riemannian surface, the curvature matrix has Pfaffian K dAK\,dA, where KK is Gaussian curvature. The theorem becomes the classical formula

12π∫MK dA=χ(M).\frac{1}{2\pi}\int_M K\,dA=\chi(M).

For the round 22-sphere, K=1K=1 and the area is 4π4\pi, so the integral gives 2=χ(S2)2=\chi(S^2).

Conventions and scope

Closedness is essential to the displayed formula. A requires a boundary transgression term. In odd dimensions the Pfaffian expression above is not present; a closed odd-dimensional manifold has Euler characteristic zero.

References
  1. Shiing-Shen Chern, “A Simple Intrinsic Proof of the Gauss–Bonnet Formula for Closed Riemannian Manifolds,” Annals of Mathematics 45 (1944), 747–752. DOI record. Relevant: §§1–3, construction of the intrinsic form and proof of the integral formula.
  2. Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: chapter 11, Euler forms and Chern–Weil theory.