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

TL;DR

This paper explores solutions to the metrizability equation in projective differential geometry, revealing how solutions stratify manifolds and generalize orbit decompositions, with implications for understanding projective compactifications.

Contribution

It extends the analysis of the metrizability equation beyond maximal rank solutions, showing stratification and geometric structures of solutions under generic conditions.

Findings

Solutions stratify the manifold by signature, generalizing Lie group orbit decompositions.

The strata have smooth structures with metrics that become singular at boundaries.

Results extend properties of normal BGG solutions to a broader class of solutions.

Abstract

Pseudo-Riemannian metrics with Levi-Civita connection in the projective class of a given torsion free affine connection can be obtained from (and 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 approach reveals and exploits interesting highly non-linear relationships between different linear geometric partial differential equations. Apart from their direct significance, the results show that, for the metrizability equation, strong results arising for so-called normal BGG solutions, and the corresponding projective holonomy reduction, extend to a far wider class of solutions. The work also provides new results for the projective compactification of scalar-flat metrics.