Metrics in Projective Differential Geometry: The Geometry of Solutions to the Metrizability Equation

Pseudo-Riemannian metrics with Levi-Civita connection in the projective class of a given torsion-free affine connection are equivalent to the maximal rank solutions of a certain overdetermined projectively invariant differential equation often called the metrizability equation. Dropping this rank assumption, we study the solutions to this equation, given less restrictive generic conditions on its prolonged system. In this setting, we find that the solution stratifies the manifold according to the strict signature (pointwise) of the solution and does this in way that locally generalizes the stratification of a model, where the model is, in each case, a corresponding Lie group orbit decomposition of the sphere. Thus the solutions give curved generalizations of such embedded orbit structures. We describe the smooth nature of the strata and determine the geometries of each of the different strata types; this includes a metric on the open strata that becomes singular at the strata boundary, with the latter a type of projective infinity for the given metric. The work also provides new results for the projective compactification of scalar-flat metrics.