Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $\pi:Q\to M$ be a smooth principal $G$-bundle over a smooth manifold $M$, and let $k\ge 1$. Let $P\in I^k(\mathfrak g)$ be an $\operatorname{Ad}$-invariant symmetric $k$-linear form on $\mathfrak g$, extended fiberwise to $\operatorname{ad}Q$-valued differential forms by the standard Chern-Weil wedge-product convention. Let $(A_t)_{t\in[0,1]}$ be a smooth path of principal connections on $Q$. For each $t\in[0,1]$, let $F_{A_t}\in\Omega^2(M;\operatorname{ad}Q)$ denote the curvature of $A_t$, and let

\begin{align*} \dot A_t:=\frac{d}{dt}A_t\in\Omega^1(M;\operatorname{ad}Q) \end{align*}

denote the tangent vector to the [affine space](/page/Affine%20Space) of connections at time $t$.

Define the Chern-Weil transgression form $T_P(A_t)\in\Omega^{2k-1}(M)$ by

\begin{align*} T_P(A_t):=k\int_{[0,1]} P(\dot A_t,F_{A_t},\ldots,F_{A_t})\,d\mathcal L^1(t), \end{align*}

where $F_{A_t}$ occurs $k-1$ times. Then

\begin{align*} dT_P(A_t)=P(F_{A_1},\ldots,F_{A_1})-P(F_{A_0},\ldots,F_{A_0}) \end{align*}

as elements of $\Omega^{2k}(M)$.