Let $M$ be a smooth manifold, let $E\to M$ be a smooth real vector bundle of finite rank $r$, and let $k\ge 0$ be an integer. For every smooth real connection $\nabla$ on $E$, let $E_{\mathbb C}:=E\otimes_{\mathbb R}\mathbb C$ be the complexification and let $\nabla^{\mathbb C}$ denote the induced complex-linear connection on $E_{\mathbb C}$. For each $j\in\{0,\dots,r\}$, let $c_j(E_{\mathbb C},\nabla^{\mathbb C})\in \Omega^{2j}(M;\mathbb C)$ denote the $j$-th Chern-Weil Chern form of the smooth complex vector bundle $E_{\mathbb C}\to M$ with connection $\nabla^{\mathbb C}$. With the conventions $c_0(E_{\mathbb C},\nabla^{\mathbb C})=1$ and $c_j(E_{\mathbb C},\nabla^{\mathbb C})=0$ for $j>r$, define the degree-$4k$ Pontryagin form by

\begin{align*} p_k(\nabla):=(-1)^k c_{2k}(E_{\mathbb C},\nabla^{\mathbb C})\in \Omega^{4k}(M;\mathbb C). \end{align*}

Then $p_k(\nabla)$ is closed. If $\nabla_0$ and $\nabla_1$ are two smooth real connections on $E$, then

\begin{align*} p_k(\nabla_0)-p_k(\nabla_1) \end{align*}

is exact. Consequently the de Rham cohomology class

\begin{align*} p_k(E):=[p_k(\nabla)]\in H^{4k}_{\mathrm{dR}}(M;\mathbb C) \end{align*}

is independent of the chosen smooth real connection $\nabla$.