Chern–Simons forms on associated bundles, and boundary terms

Let E be a principle bundle over a compact manifold M with compact structural group G. For any G-invariant polynomial P, the transgressive forms \(TP(\omega)\) defined by Chern and Simons in (Ann. Math. 99:48–69, 1974) are shown to extend to forms \(\Phi P(\omega)\) on associated bundles B with fiber a quotient F = G/H of the group. These forms satisfy a heterotic formula \(d\Phi P(\omega)=P(\Omega)-P(\Psi),\) relating the characteristic form \(P(\Omega)\) to a fiber-curvature characteristic form. For certain natural bundles B, \(P(\Psi)=0\) , giving a true transgressive form on the associated bundle, which leads to the standard obstruction properties of characteristic classes as well as natural expressions for boundary terms. These forms also yield new secondary characteristic classes giving refined information about the associated bundles B.