= Projection connection
{title2=$\nabla=p\,d,\qquad F=p(dp)^2$}
A smooth self-adjoint <projection> $p:M\to M_N(\mathbb C)$ of locally constant rank defines the <Hermitian vector bundle> $\operatorname{im}p$ inside the trivial bundle. Projecting the flat derivative gives the <unitary connection> $\nabla s=p\,ds$ for $ps=s$. Differentiating $p^2=p$ gives $p\,dp=dp(1-p)$ and $p\,dp\,p=0$. For a section $s$, $\nabla^2s=p\,dp\wedge ds=p(dp)^2s$, proving the curvature formula. Relative to $\operatorname{im}p\oplus\ker p$, $dp$ is off-diagonal and $(dp)^2$ is diagonal. Therefore $d\operatorname{tr}(p(dp)^{2k})=\operatorname{tr}(dp)^{2k+1}=0$. These are explicit closed representatives of the <Chern character>.
Back to article page
