Chern–Gauss–Bonnet theorem

Remarks

Chern gave an intrinsic proof in 1944: on the unit sphere bundle of $M$, the form $\operatorname{Pf}(\Omega)$ pulled back is exact, $d\Pi$, by a transgression formula. Take a vector field $X$ with isolated zeros, and integrate over $M$ minus small balls around the zeros. Stokes' theorem gives the sum of the indices of $X$, which is $\chi(M)$ by the Poincaré–Hopf theorem.

Example: for $m = 1$, $\operatorname{Pf}(\Omega) = \Omega_{12} = K \, dA$ with $K$ the Gaussian curvature, and the formula is the Gauss–Bonnet theorem $\int_M K \, dA = 2\pi \chi(M)$. For $m = 2$ it reads $$\chi(M) = \frac{1}{32\pi^2} \int_M \bigl(\abs{\operatorname{Rm}}^2 - 4 \abs{\operatorname{Ric}}^2 + R^2\bigr) \, dV,$$ with $\operatorname{Rm}$ the Riemann tensor, $\operatorname{Ric}$ the Ricci tensor and $R$ the scalar curvature. For the unit sphere $S^4$ the integrand is $24 - 144 + 144 = 24$ and the volume is $8\pi^2/3$, which gives $\chi(S^4) = 2$.

For odd $n$ the Euler characteristic of a closed manifold is $0$, so there is no analogous curvature formula.

Going deeper

For a student

The theorem generalizes the Gauss–Bonnet theorem for surfaces to all even dimensions. In the language of characteristic classes, $\operatorname{Pf}(\Omega / 2\pi)$ represents the Euler class of the tangent bundle in de Rham cohomology. The theorem then says that the Euler number of the tangent bundle is $\chi(M)$. It is a model case of Chern–Weil theory and a special case of the Atiyah–Singer index theorem, for the operator $d + \delta$ from even to odd forms.