Euler class via Chern–Weil theory

Let $M$ be a smooth manifold and let $\pi:E\to M$ be an oriented real vector bundle of rank $2m$. Choose a Euclidean metric on $E$ and a compatible connection $\nabla$ whose structure group is reduced to $SO(2m)$. Let $F_\nabla\in\Omega^2(M;\mathfrak{so}(E))$ be its curvature .

Definition (Euler form and Euler class)

The Pfaffian $\mathrm{Pf}$ is an $Ad$-invariant polynomial of degree $m$ on $\mathfrak{so}(2m)$ characterized by $\mathrm{Pf}(A)^2=\det(A)$ for skew-symmetric matrices $A$. The Euler form of $\nabla$ is the closed $2m$-form

(Here $\mathrm{Pf}$ is applied fiberwise after choosing a local oriented orthonormal frame.)

Then:

1. $d\,e(\nabla)=0$, where $d$ is the exterior derivative .
2. The de Rham class $[e(\nabla)]\in H^{2m}_{\mathrm{dR}}(M)$ is independent of the choice of metric-compatible oriented connection.
3. The Euler class $e(E)\in H^{2m}(M;\mathbb Z)$ is the unique integral class mapping to $[e(\nabla)]$ under the natural map to real (or de Rham) cohomology.

Naturality holds: for any smooth map $f:N\to M$,

Examples

1. Oriented rank-2 bundles. If $\mathrm{rank}(E)=2$, then $F_\nabla$ is an $\mathfrak{so}(2)\cong\mathbb R$-valued 2-form. In an oriented orthonormal frame, $F_\nabla$ is represented by a scalar 2-form $\Omega$, and

   $$e(\nabla)=\frac{1}{2\pi}\Omega.$$

   In particular, the Euler class is represented by $\frac{1}{2\pi}\Omega$.

2. Tangent bundle of an oriented closed surface. For a closed oriented surface $\Sigma$, the Euler class of $T\Sigma$ satisfies

   $$\langle e(T\Sigma),[\Sigma]\rangle=\chi(\Sigma),$$

   and $e(\nabla)$ is the Gauss curvature form normalized by $2\pi$ for the Levi-Civita connection.

3. Flat oriented bundles. If $\nabla$ is flat ($F_\nabla=0$), then $e(\nabla)=0$, so $e(E)$ vanishes in real cohomology (and hence in rational cohomology).