Characteristic forms represent topological characteristic classes over the reals

Prove componentwise: every component is open in a manifold chart and inherits the stated smooth scope; its singular cochains are the product across components, and full AC supplies componentwise cocycles and primitives, so equality on components is equality on $M$; the same componentwise reasoning applies to each smooth flag space below. By [F5], the topological classes in [F7]–[F11] are defined, and [F6] gives the real de Rham ring isomorphism. If $M=\mathrm{\varnothing }$, all singular and de Rham groups are zero, so the assertions hold there as well.

For a complex bundle $E$ of rank $r$ with Hermitian connection $\mathrm{\nabla }$, take on each component the complex flag projection $q:F(E)\to M$ from [F3]; it has injective real-cohomology pullback and $q^{\ast }E=L_1\oplus \cdots \oplus L_r$ orthogonally. The flag space is in the smooth scope of [F5], ranks $0,1$ use the identity map, and $q^{\ast }\mathrm{\nabla }$ is Hermitian.

Let $V$ be a real rank-$r$ bundle with arbitrary real connection $D$. For a real matrix $A$, $P_j(A)=[t^{2j}]\det (I+tA\mathrm{/}(2\pi ))$ is a real ${\mathrm{GL}}_r(\mathbb{R})$-invariant polynomial. Writing $e_k(A)=[t^k]\det (I+tA)$, the complexified-curvature formula [F1] gives $c_k(D_{\mathbb{C}})=i^k{e}_k({\mathrm{\Omega }}_D)\mathrm{/}(2\pi {)}^k$, hence as actual real forms $p_j(D)=(-1{)}^j{c}_{2j}(D_{\mathbb{C}})=P_j({\mathrm{\Omega }}_D)$. Chern–Weil connection independence [F2] for the real general-linear reduction compares all real connections in real de Rham cohomology.

On one oriented two-plane summand, take a positive orthonormal frame $(e_1,e_2)$ with $J{e}_1=e_2$, where $J$ is its orientation complex structure. Metric compatibility gives $\mathrm{\nabla }{e}_1=\alpha e_2$ and $\mathrm{\nabla }{e}_2=-\alpha e_1$, so the curvature matrix is $\left(\begin{array}{cc}0& -d\alpha \\ d\alpha & 0\end{array}\right)$ and the library Pfaffian convention [F1] gives $-d\alpha \mathrm{/}(2\pi )$. The associated complex line with induced Hermitian metric has connection form $i\alpha$, curvature $id\alpha$, and first Chern form $-id\alpha \mathrm{/}(2\pi i)=-d\alpha \mathrm{/}(2\pi )$. By [F14] and [F8], its real class is the coefficient image of the Thom-normalized Euler class of the plane.

Let $P_a$ be the smooth orthogonal projection onto $L_a$ and set ${\mathrm{\nabla }}^{a}s=P_a((q^{\ast }\mathrm{\nabla })s)$ for sections of $L_a$. Since $P_a(fs)=f{P}_a(s)$, the projected operator obeys the connection Leibniz rule; since $⟨P_{a}u,t⟩=⟨u,t⟩$ for $t\in L_a$, the Hermitian metric identity for $q^{\ast }\mathrm{\nabla }$ restricts to the same identity after projection. Thus ${\mathrm{\nabla }}^{\oplus }={⨁}_a{\mathrm{\nabla }}^a$ is Hermitian on the same complex bundle as $q^{\ast }\mathrm{\nabla }$. By [F2], connection independence and naturality identify the de Rham classes of $c_j(q^{\ast }\mathrm{\nabla })$ and $q^{\ast }{c}_j(\mathrm{\nabla })$ with those of $c_j({\mathrm{\nabla }}^{\oplus })$ and $q^{\ast }[c_j(\mathrm{\nabla })]$, respectively.

The curvature of ${\mathrm{\nabla }}^{\oplus }$ is block diagonal, so [F1] gives $c({\mathrm{\nabla }}^{\oplus })={\bigwedge }_{a=1}^r(1+c_1({\mathrm{\nabla }}^a))$. For each line, [F8] and [F14] give $J_{F(E)}([c_1({\mathrm{\nabla }}^a)])={\rho }_{F(E)}(c_1(L_a))$; multiplicativity of [F6] and the topological Whitney formula [F7] then give $J_{F(E)}([c_j({\mathrm{\nabla }}^{\oplus })])={\rho }_{F(E)}(c_j(q^{\ast }E))={\rho }_{F(E)}(q^{\ast }{c}_j(E))$ for every $j$. Naturality in [F2] and [F6] makes the pullback of $J_M([c_j(\mathrm{\nabla })])-{\rho }_M(c_j(E))$ zero. Injectivity in [F3] proves the complex assertion.

Let $W$ be an oriented Euclidean bundle of rank $2m$ with metric-compatible connection $\mathrm{\nabla }$. The real flag projection [F4] gives $q:F(W)\to M$ with injective pullback and an ordered orthogonal splitting $q^{\ast }W=L_1\oplus \cdots \oplus L_m$ into oriented two-planes. Pull back $\mathrm{\nabla }$ and project orthogonally to each summand; the Euclidean metric identity restricts under orthogonal projection by $⟨P_{a}u,t⟩=⟨u,t⟩$, so the projected connections and their direct sum are metric-compatible on the same oriented Euclidean bundle as $q^{\ast }\mathrm{\nabla }$. This is the same projection calculation as in step 2.1. By [F2], the Pfaffian classes of these two connections agree.

By [A1] and [F12], choose a Euclidean metric on $V$ and a compatible connection $D_g$; its complexification is Hermitian for the induced metric. The complex result in step 3.1 identifies $J_M([c_{2j}((D_g{)}_{\mathbb{C}})])$ with ${\rho }_M(c_{2j}(V_{\mathbb{C}}))$. Multiplying by $(-1{)}^j$ and using [F9] and step 1.3 proves the Pontryagin equality for $D_g$, while step 1.3 permits replacing $D$ by $D_g$ in the Pontryagin de Rham class. If $2j>r$, both sides vanish by [F1] and [F9]; if $j=0$, both are the unit.

The Pfaffian of the block-diagonal curvature in step 3.2 is the product of the rank-two Pfaffians. Step 1.4, the ring isomorphism [F6], and the Euler Whitney product [F10] give $J_{F(W)}([e(q^{\ast }\mathrm{\nabla })])={\rho }_{F(W)}(e(q^{\ast }W))$. Naturality of the Euler class [F10] and characteristic form [F2] identifies this with the pullback of the difference on $M$; injectivity in [F4] proves the Euler assertion.

The degree-zero classes in the total forms are units; rank-zero Chern and Pontryagin bundles have no positive-degree coefficients, and a rank-zero oriented Euclidean bundle has Euler form and class equal to the unit by [F1] and [F11]. A complex line is covered by steps 1.2, 2.1, and 3.1; the rank-two Euler sign by step 1.4; and the rank cutoffs for $c_j,p_j$ by [F1], [F9], and step 4.1. Boundary points are covered by the half-space flag, connection, pullback, and de Rham suppliers [F2]–[F6]. There is no odd-rank Euler-form clause or if-and-only-if assertion. Full AC is used in step 1.1 for componentwise cohomology, through the flag and characteristic-class/Thom/Euler suppliers [F3]–[F5], [F7]–[F11], and in step 4.1 for compatible-connection existence; the curvature algebra and supplied-connection comparisons are choice-free.