Let $G$ be a Lie group with identity element $e$ and [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $M$ be a smooth manifold, and let $\pi:P\to M$ be a smooth principal right $G$-bundle with right action $R_g:P\to P$. Let $A\in\Omega^1(P;\mathfrak g)$ be a principal connection form with curvature
Let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over $\mathbb K$, where $\mathbb K\in\{\mathbb R,\mathbb C\}$, and let $\rho:G\to GL(V)$ be a smooth representation. Let
where $(pg,v)\sim(p,\rho(g)v)$ for $p\in P$, $g\in G$, and $v\in V$. Equip $E$ with the vector bundle connection $\nabla^E$ induced by the horizontal distribution $\ker A\subset TP$, and let $F_E$ denote its curvature.
paragraph
admin
For every [open set](/page/Open%20Set) $U\subset M$ and every smooth local section $s:U\to P$, define
Consequently, let $k\ge 0$, and let $q:\mathfrak{gl}(V)^k\to \mathbb K$ be a $GL(V)$-invariant symmetric $k$-linear polynomial, with the convention that $k=0$ means a constant element of $\mathbb K$. Here $GL(V)$ acts on $\mathfrak{gl}(V)$ by conjugation, so $q$ is invariant when
Let $\operatorname{Ad}:G\to GL(\mathfrak g)$ denote the adjoint representation of $G$ on $\mathfrak g$. Then $\rho^*q$ is invariant under $\operatorname{Ad}$, and the Chern-Weil forms satisfy locally