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.

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

e(\nabla)\;:=\;\mathrm{Pf}\!\Big(\frac{1}{2\pi}F_\nabla\Big)\in\Omega^{2m}(M).

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

Then:

$d\,e(\nabla)=0$ , where $d$ is the exterior derivative.
The de Rham class $[e(\nabla)]\in H^{2m}_{\mathrm{dR}}(M)$ is independent of the choice of metric-compatible oriented connection.
The Euler class $e(E)\in H^{2m}(M;\mathbb Z)$ is the canonical integral class whose image under the natural map to real (or de Rham) cohomology is $[e(\nabla)]$ . Its real image does not by itself determine the integral class when cohomology has torsion.

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

e(f^*E)=f^*e(E).

Examples

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 $\frac{1}{2\pi}\Omega$ .

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, for a Riemannian metric and its Levi-Civita connection, $e(\nabla)=K\,\operatorname{vol}_g/(2\pi)$ , where $K$ is the Gauss curvature and $\operatorname{vol}_g$ is the oriented volume form.

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).