Let $\pi:P\to M$ be a smooth principal $G$-bundle over a Riemannian manifold $(M,g)$, let $\mathfrak g$ be the Lie algebra of $G$, and let $(\cdot,\cdot)_{\mathfrak g}$ be an $\operatorname{Ad}$-invariant [inner product](/page/Inner%20Product) on $\mathfrak g$, meaning that

\begin{align*} (\operatorname{Ad}_a \xi,\operatorname{Ad}_a \eta)_{\mathfrak g}=(\xi,\eta)_{\mathfrak g} \end{align*}

for every $a\in G$ and every $\xi,\eta\in\mathfrak g$.

Let $\omega\in\Omega^1(P;\mathfrak g)$ be a connection form with curvature $\Omega\in\Omega^2(P;\mathfrak g)$, and let $F_A\in\Omega^2(M;\operatorname{Ad}(P))$ be the associated adjoint-bundle-valued curvature form. Suppose $s_i:U_i\to P$ and $s_j:U_j\to P$ are smooth local sections over open sets $U_i,U_j\subset M$, and suppose that on $U_i\cap U_j$ there is a smooth transition map $g_{ij}:U_i\cap U_j\to G$ such that

\begin{align*} s_j(x)=s_i(x)g_{ij}(x) \end{align*}

for every $x\in U_i\cap U_j$. Define the local curvature representatives

\begin{align*} F_i:=s_i^*\Omega\in\Omega^2(U_i;\mathfrak g),\qquad F_j:=s_j^*\Omega\in\Omega^2(U_j;\mathfrak g). \end{align*}

If on $U_i\cap U_j$ these representatives satisfy the gauge transformation law

\begin{align*} F_j=\operatorname{Ad}_{g_{ij}^{-1}}F_i, \end{align*}

then for every $x\in U_i\cap U_j$,

\begin{align*} |F_j(x)|^2=|F_i(x)|^2, \end{align*}

where the pointwise norm is induced by the Riemannian metric $g$ on $\Lambda^2T^*M$ and by $(\cdot,\cdot)_{\mathfrak g}$ on $\mathfrak g$. Consequently the local densities $|F_i|^2\,d\operatorname{vol}_g$ glue to a globally defined density $|F_A|^2\,d\operatorname{vol}_g$ on $M$, and the Yang-Mills energy

\begin{align*} \operatorname{YM}(A):=\int_M |F_A(x)|^2\,d\operatorname{vol}_g(x) \end{align*}

is independent of the chosen local sections and trivializations whenever the integral is defined.