  We approach the problem of finding obstructions to curvature distinguished
Riemannian metrics by considering Lorentzian metrics to which they are dual in
a suitable sense. Obstructions to the latter then yield obstructions to the
former. This framework applies both locally and globally, including to compact
manifolds, and is sensitive to various aspects of curvature. Here we apply it
in two different ways. First, by embedding a Riemannian manifold into a
Lorentzian one and utilizing Penrose's "plane wave limit," we find necessary
local conditions, in terms of the Hessian of just one function, for large
classes of Riemannian metrics to contain within them those that have parallel
Ricci tensor, or are Ricci-flat, or are locally symmetric. Second, by
considering Riemannian metrics dual to constant curvature Lorentzian metrics
via a type of Wick rotation, we are able to rule out the existence of a family
of compact Riemannian manifolds (in all dimensions) that deviate from constant
curvature in a precise sense.
