[proofplan] The proof is the standard Chern-Weil construction for the Pfaffian invariant polynomial. First we check that the normalized Pfaffian of the curvature is globally defined by using the $SO(2m)$-congruence invariance of the Pfaffian. The [Chern-Weil closedness theorem](/theorems/7039) then gives closedness, and the Chern-Weil transgression formula applied to the affine path between two metric connections gives connection independence. Finally, the sign and normalization are fixed by the rank-two convention, and the standard Thom-class characterization of the Euler class identifies the resulting de Rham class with the real image of the integral Euler class. [/proofplan] [step:Use oriented orthonormal frames to define the Pfaffian curvature form globally] Let $SO(2m)$ denote the group of orientation-preserving orthogonal automorphisms of $\mathbb R^{2m}$, and let $\mathfrak{so}(2m)$ denote its [Lie algebra](/page/Lie%20Algebra), equivalently the [vector space](/page/Vector%20Space) of real skew-symmetric $2m\times 2m$ matrices. For $m=0$, $SO(0)$ is the trivial group, $\mathfrak{so}(0)=\{0\}$, and the Pfaffian of the unique $0\times 0$ skew-symmetric matrix is $1$. Let $U\subset M$ be an [open set](/page/Open%20Set) over which $E$ admits a positive oriented $h$-orthonormal frame \begin{align*} s=(s_1,\dots,s_{2m}). \end{align*} In this frame the curvature of the $h$-compatible connection $\nabla$ is represented by a skew-symmetric matrix of $2$-forms \begin{align*} \Omega_s=(\Omega_{ij})_{1\le i,j\le 2m}\in \Omega^2(U;\mathfrak{so}(2m)). \end{align*} Define the local form \begin{align*} e_s(\nabla):=\operatorname{Pf}\left(\frac{\Omega_s}{2\pi}\right)\in \Omega^{2m}(U). \end{align*} If $t=(t_1,\dots,t_{2m})$ is another positive oriented $h$-orthonormal frame on an open set $V\subset M$, then on $U\cap V$ there is a smooth transition map \begin{align*} g:U\cap V\to SO(2m) \end{align*} such that $t=s g$. The curvature matrices satisfy \begin{align*} \Omega_t=g^{-1}\Omega_s g. \end{align*} By the Pfaffian congruence identity and the determinant identity for skew-symmetric matrices, as in [citetheorem:9773], one has \begin{align*} \operatorname{Pf}(g^{-1}Ag)=\det(g)^{-1}\operatorname{Pf}(A) \end{align*} for every $A\in \mathfrak{so}(2m)$. Since $g(x)\in SO(2m)$ for every $x\in U\cap V$, $\det(g(x))=1$. Hence \begin{align*} \operatorname{Pf}\left(\frac{\Omega_t}{2\pi}\right)=\operatorname{Pf}\left(\frac{\Omega_s}{2\pi}\right) \end{align*} on $U\cap V$. Therefore the local forms $e_s(\nabla)$ glue to a well-defined global form \begin{align*} e(\nabla)\in \Omega^{2m}(M). \end{align*} [guided] Let $SO(2m)$ denote the group of orientation-preserving orthogonal automorphisms of $\mathbb R^{2m}$, and let $\mathfrak{so}(2m)$ denote its Lie algebra, equivalently the vector space of real skew-symmetric $2m\times 2m$ matrices. For $m=0$, $SO(0)$ is the trivial group, $\mathfrak{so}(0)=\{0\}$, and the Pfaffian of the unique $0\times 0$ skew-symmetric matrix is $1$. We first have to justify that the displayed formula for $e(\nabla)$ does not depend on a local frame. Let $U\subset M$ be an open set on which $E$ has a positive oriented $h$-orthonormal frame \begin{align*} s=(s_1,\dots,s_{2m}). \end{align*} Because $\nabla$ is compatible with $h$, its connection matrix in this frame is $\mathfrak{so}(2m)$-valued, and therefore its curvature matrix is a skew-symmetric matrix of $2$-forms \begin{align*} \Omega_s=(\Omega_{ij})_{1\le i,j\le 2m}\in \Omega^2(U;\mathfrak{so}(2m)). \end{align*} The local Euler form is defined by \begin{align*} e_s(\nabla):=\operatorname{Pf}\left(\frac{\Omega_s}{2\pi}\right)\in \Omega^{2m}(U). \end{align*} Now suppose $t=(t_1,\dots,t_{2m})$ is another positive oriented $h$-orthonormal frame over an open set $V\subset M$. On $U\cap V$, the change of frame is a smooth map \begin{align*} g:U\cap V\to SO(2m) \end{align*} defined by $t=s g$. The positivity of the two orientations is exactly what puts $g$ in $SO(2m)$ rather than $O(2m)$. Curvature transforms by conjugation under change of frame, so \begin{align*} \Omega_t=g^{-1}\Omega_s g. \end{align*} The Pfaffian is not invariant under every congruence without a determinant factor; the precise identity, proved in [citetheorem:9773], is \begin{align*} \operatorname{Pf}(g^{-1}Ag)=\det(g)^{-1}\operatorname{Pf}(A) \end{align*} for skew-symmetric matrices $A$. Since $g(x)\in SO(2m)$, we have $\det(g(x))=1$ for every $x\in U\cap V$. Substituting $A=\Omega_s/(2\pi)$ gives \begin{align*} \operatorname{Pf}\left(\frac{\Omega_t}{2\pi}\right)=\operatorname{Pf}\left(\frac{\Omega_s}{2\pi}\right). \end{align*} Thus the local formulas agree on overlaps. This proves that they glue to a globally defined differential form \begin{align*} e(\nabla)\in \Omega^{2m}(M). \end{align*} [/guided] [/step] [step:Apply Chern-Weil closedness to the Pfaffian invariant polynomial] The Pfaffian \begin{align*} \operatorname{Pf}:\mathfrak{so}(2m)\to \mathbb R \end{align*} is a [homogeneous polynomial](/page/Homogeneous%20Polynomial) of degree $m$. The previous step shows that it is invariant under the adjoint action of $SO(2m)$. Since $\nabla$ is an $h$-compatible connection, it is induced by a principal connection on the positive [oriented orthonormal frame bundle](/theorems/6245) $P_{SO}(E,h)\to M$, and $F_\nabla$ is the curvature of the associated vector bundle connection. The hypotheses of the Chern-Weil closedness theorem are therefore satisfied by the principal $SO(2m)$-bundle $P_{SO}(E,h)$ and the invariant polynomial \begin{align*} X\mapsto \operatorname{Pf}\left(\frac{X}{2\pi}\right). \end{align*} By [citetheorem:9757], the form \begin{align*} e(\nabla)=\operatorname{Pf}\left(\frac{F_\nabla}{2\pi}\right) \end{align*} is closed. [/step] [step:Use Chern-Weil transgression to prove independence of the metric connection] If $m=0$, then $E$ has rank $0$ and $F_\nabla$ is represented in the unique oriented orthonormal frame by the unique $0\times 0$ skew-symmetric matrix. By the stated Pfaffian convention, $e(\nabla)=1\in\Omega^0(M)$ for every $h$-compatible connection $\nabla$, so both closedness and connection independence are immediate in this case. Assume now that $m\ge 1$. Let $\nabla_0$ and $\nabla_1$ be $h$-compatible connections on $E$. Define the affine path of connections \begin{align*} \nabla_t:=(1-t)\nabla_0+t\nabla_1 \end{align*} for $t\in[0,1]$. Because compatibility with $h$ is an affine condition on the connection, each $\nabla_t$ is again $h$-compatible. Let \begin{align*} A:=\nabla_1-\nabla_0\in \Omega^1(M;\mathfrak{so}(E,h)) \end{align*} be the difference one-form, and let \begin{align*} F_t\in \Omega^2(M;\mathfrak{so}(E,h)) \end{align*} be the curvature of $\nabla_t$. Apply the Chern-Weil transgression formula to the invariant polynomial \begin{align*} P:\mathfrak{so}(2m)\to \mathbb R,\qquad P(X)=\operatorname{Pf}\left(\frac{X}{2\pi}\right). \end{align*} The hypotheses are satisfied because $P$ is $SO(2m)$-invariant and homogeneous of degree $m$, and because $(\nabla_t)_{t\in[0,1]}$ is a smooth path of connections induced by a smooth path of principal connections on the same principal bundle $P_{SO}(E,h)\to M$. By [citetheorem:9791], there is a transgression form \begin{align*} T_P(\nabla_t)\in \Omega^{2m-1}(M) \end{align*} such that \begin{align*} dT_P(\nabla_t)=P(F_1)-P(F_0). \end{align*} Since $P(F_i)=e(\nabla_i)$ for $i\in\{0,1\}$, this gives \begin{align*} e(\nabla_1)-e(\nabla_0)=dT_P(\nabla_t). \end{align*} Thus $e(\nabla_1)-e(\nabla_0)$ is exact, and therefore \begin{align*} [e(\nabla_0)]=[e(\nabla_1)]\in H^{2m}_{\mathrm{dR}}(M). \end{align*} [/step] [step:Identify the normalized Pfaffian class with the real Euler class] It remains to identify the common de Rham class. We invoke the Chern-Weil comparison theorem for the Euler class: for an oriented real vector bundle of even rank equipped with a bundle metric and a compatible metric connection, the de Rham class of the normalized Pfaffian curvature form equals the image of the integral Euler class under the coefficient map and the de Rham comparison isomorphism, provided the Pfaffian convention is the one determined by the Thom-class orientation normalization. The hypotheses are satisfied here because $E\to M$ is oriented, has even rank $2m$, carries the bundle metric $h$, and $\nabla$ is $h$-compatible. The sign convention also matches the theorem's normalization: in a positive oriented orthonormal frame for an oriented rank-two bundle, a curvature matrix with entry $F_{12}$ gives \begin{align*} e(\nabla)=\frac{F_{12}}{2\pi}. \end{align*} With this convention, the comparison theorem gives \begin{align*} [e(\nabla)]=e(E)_{\mathbb R}\in H^{2m}_{\mathrm{dR}}(M). \end{align*} Equivalently, if \begin{align*} H^{2m}(M;\mathbb Z)\to H^{2m}(M;\mathbb R)\cong H^{2m}_{\mathrm{dR}}(M) \end{align*} denotes the coefficient map followed by the de Rham comparison isomorphism, then the image of $e(E)\in H^{2m}(M;\mathbb Z)$ is $[e(\nabla)]$. This proves closedness, independence of the compatible metric connection, and the claimed identification with the integral Euler class. [/step]