Let $M$ be a paracompact smooth manifold, let $m\ge 0$, and let $E\to M$ be an oriented smooth real vector bundle of rank $2m$ equipped with a bundle metric $h$. Let $P_{SO}(E,h)\to M$ be the principal $SO(2m)$-bundle of positive oriented $h$-orthonormal frames. Let $SO(2m)$ denote the group of orientation-preserving orthogonal automorphisms of $\mathbb R^{2m}$, with [Lie algebra](/page/Lie%20Algebra) $\mathfrak{so}(2m)$ the skew-symmetric endomorphisms of $\mathbb R^{2m}$; for $m=0$ these are the trivial group and the zero Lie algebra. Let $\nabla$ be an $h$-compatible connection on $E$, equivalently the connection induced from a principal connection on $P_{SO}(E,h)$, and let

\begin{align*} F_\nabla\in \Omega^2(M;\mathfrak{so}(E,h)) \end{align*}

be its curvature, where $\mathfrak{so}(E,h)\to M$ is the bundle of $h$-skew-adjoint endomorphisms of $E$. Define the Euler form of $\nabla$ by the normalized Pfaffian convention

\begin{align*} e(\nabla):=\operatorname{Pf}\left(\frac{F_\nabla}{2\pi}\right)\in \Omega^{2m}(M), \end{align*}

where the Pfaffian is evaluated in positive oriented local $h$-orthonormal frames, $\operatorname{Pf}$ of the unique $0\times 0$ skew-symmetric matrix is $1$, and the sign is fixed so that, in rank $2$, a curvature matrix with positive oriented entry $F_{12}$ gives $e(\nabla)=F_{12}/(2\pi)$.

Then $e(\nabla)$ is closed. If $\nabla_0$ and $\nabla_1$ are two $h$-compatible connections on $E$, then

\begin{align*} [e(\nabla_0)]=[e(\nabla_1)]\in H^{2m}_{\mathrm{dR}}(M). \end{align*}

Moreover, under the coefficient homomorphism

\begin{align*} H^{2m}(M;\mathbb Z)\to H^{2m}(M;\mathbb R) \end{align*}

and the de Rham comparison isomorphism

\begin{align*} H^{2m}(M;\mathbb R)\cong H^{2m}_{\mathrm{dR}}(M), \end{align*}

this common de Rham class is the image of the integral Euler class $e(E)\in H^{2m}(M;\mathbb Z)$. Equivalently,

\begin{align*} [e(\nabla)]=e(E)_{\mathbb R}\in H^{2m}_{\mathrm{dR}}(M), \end{align*}

where $e(E)_{\mathbb R}$ denotes the real de Rham image of $e(E)$.