The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can place restrictions on the manifold.
What pointwise meaning can one ascribe to the Gauss-Bonnet integrand? What ramifications can one expect if one finds a “higher Einstein metric”? That is, one where the Pfaffian of the curvature equals the volume form (upto a constant).
In a related vein, on a Kahler manifold, the first Chern form being the Ricci form has local meaning (and was profitably used to prove that Fano manifolds are simply connected for instance). Do the higher Chern forms have pointwise meaning?