= Pontryagin class
{c}
{title2=$p_k(E)=(-1)^kc_{2k}(E\otimes_{\mathbb R}\mathbb C)$}
For a real <vector bundle> $E$, its Pontryagin classes are the even-index <Chern classes> of its complexification with the displayed alternating signs. They lie in integral <cohomology> of degrees $4k$ and obey a total-class product formula under <direct sums>. If a metric <connection on a vector bundle> has curvature matrix $R$, its first real representative is $p_1(E)=-\operatorname{tr}(R\wedge R)/(8\pi^2)$. The <Bianchi identity> makes this representative closed, and <Chern-Weil connection transgression> makes its <de Rham cohomology> class independent of the connection. This normalization gives degree-four <A-hat form> $-p_1/24$.
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