Projection connection ($\nabla =pd,F=p(dp{)}^2$)

A smooth self-adjoint projection $p:M\to M_N(\mathbb{C})$ of locally constant rank defines the Hermitian vector bundle $\mathrm{im}p$ inside the trivial bundle. Projecting the flat derivative gives the unitary connection $\nabla s=pds$ for $ps=s$. Differentiating $p^2=p$ gives $pdp=dp(1-p)$ and $pdpp=0$. For a section $s$, ${\nabla }^{2}s=pdp\wedge ds=p(dp{)}^{2}s$, proving the curvature formula. Relative to $\mathrm{im}p\oplus \ker p$, $dp$ is off-diagonal and $(dp{)}^2$ is diagonal. Therefore $d\mathrm{tr}(p(dp{)}^{2k})=\mathrm{tr}(dp{)}^{2k+1}=0$. These are explicit closed representatives of the Chern character.