The Chern–Weil homomorphism and invariant polynomials

Let $P$ be a symmetric $k$-linear form on the Lie algebra invariant under the adjoint action. Evaluating on $k$ curvature factors gives a degree-$2k$ form. The principal curvature is horizontal and equivariant; applying $P$ makes it invariant as well, so it is basic and descends uniquely to the base. In local notation this says $P(F_j,\ldots,F_j)=P(F_i,\ldots,F_i)$ because curvature changes by conjugation. An arbitrary polynomial lacking this invariance need not produce an atlas-independent expression.

The Bianchi identity and infinitesimal invariance imply that the descended form is closed. Varying the connection along an affine path shows that its derivative is exact, so the de Rham class depends on the bundle and $P$, rather than on the chosen connection. Multiplication of invariant polynomials becomes wedge product of cohomology classes, yielding the Chern–Weil homomorphism. The exact transgression formula will be derived next; here it justifies why a continuously changing curvature density can represent an unchanged class.

For a unitary connection with skew-Hermitian matrix potential and $\nabla=d+A$, use the total Chern form $\det(I+iF/(2\pi))$. Thus the first Chern form is $i\operatorname{Tr}F/(2\pi)$. This equals the coefficient convention in $\det(\lambda I-F/(2\pi i))$; forgetting the minus sign in that expression reverses the first class. For a real orthogonal connection the normalized first Pontryagin form is $-\operatorname{Tr}(F\wedge F)/(8\pi^2)$. The representation used for the trace is the defining real representation.

For an oriented real rank-$2m$ bundle with an orthogonal connection, the Euler form is the Pfaffian of $F/(2\pi)$. Fix the standard convention that the Pfaffian of $\begin{pmatrix}0&b\\-b&0\end{pmatrix}$ equals $b$. For the oriented plane with complex structure $J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, a potential $A=J\alpha$ has Euler form $-d\alpha/(2\pi)$. Under the corresponding complex line convention this agrees with the first Chern form, since its complex curvature is $i\,d\alpha$.

A compact oriented base allows integration of top-degree characteristic forms to obtain real characteristic numbers. When the polynomial is integrally normalized, these agree with the corresponding integer cohomological pairings. A local calculation alone does not establish integrality; that statement uses the topological normalization and the bundle's global gluing. An arbitrary real multiple of an invariant polynomial still defines a valid real Chern–Weil class, but its periods need not be integers.

A flat connection makes every positive-degree curvature polynomial vanish, so the corresponding real characteristic classes vanish. Integral torsion classes can nevertheless remain nonzero, and flat holonomy can retain further information. Likewise two different connections may have different local curvature densities while their Chern–Weil forms differ only by an exact form. On a closed base their matching top-degree integrals then agree by Stokes. With a boundary, an exact difference can produce a boundary integral, so closed-base and relative-boundary conclusions must be distinguished.