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Pseudo-Riemannian metrics with Levi-Civita connection in the projective classof a given torsion free affine connection can be obtained from (and areequivalent to) the maximal rank solutions of a certain overdeterminedprojectively invariant differential equation often called the metrizabilityequation. Dropping this rank assumption we study the solutions to this equationgiven less restrictive generic conditions on its prolonged system. In thissetting we find that the solution stratifies the manifold according to thestrict signature (pointwise) of the solution and does this in way that locallygeneralizes the stratification of a model, where the model is, in each case, acorresponding Lie group orbit decomposition of the sphere. Thus the solutionsgive curved generalizations of such embedded orbit structures. We describe thesmooth nature of the strata and determine the geometries of each of thedifferent strata types; this includes a metric on the open strata that becomessingular at the strata boundary, with the latter a type of projective infinityfor the given metric. The approach reveals and exploits interesting highlynon-linear relationships between different linear geometric partialdifferential equations. Apart from their direct significance, the results showthat, for the metrizability equation, strong results arising for so-callednormal BGG solutions, and the corresponding projective holonomy reduction,extend to a far wider class of solutions. The work also provides new resultsfor the projective compactification of scalar-flat metrics.

Metrics in projective differential geometry: the geometry of solutions to the metrizability equation