Pontrjagin Forms And Invariant Objects Related To The Q-Curvature

It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfa. an. Working on even-dimensional manifolds, we show how the k-form operators Q(k) of [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfa. an, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qk give rise to conformally invariant quadratic forms circle minus(k) on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfa. an (at k = 0). Using a different construction, we show that the Qk operators yield a map from conformal structures to Lagrangian subspaces of the direct sum H-k circle plus H-k (where H-k is the dual of the de Rham cohomology space H-k); in an appropriate sense this generalizes the period map. We couple the Qk operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Q(k)-operators the Q-curvature prescription problem.