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

Obstructions to distinguished Riemannian metrics via Lorentzian geometry