2018/02/17 by Keegan J. Flood, Flood, Keegan J., A. Rod Gover +1 · 1 citation
Mathematics · Physics and Astronomy · #53A30 #53B10 #53C21 #58J60 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Primary 53A20 #Secondary 35N10
paper · pdf · doi:10.48550/arxiv.1802.06329
openalex publication_date 2018/02/17 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Pseudo-Riemannian metrics with Levi-Civita connection in the projective class\nof a given torsion free affine connection can be obtained from (and are\nequivalent to) the maximal rank solutions of a certain overdetermined\nprojectively invariant differential equation often called the metrizability\nequation. Dropping this rank assumption we study the solutions to this equation\ngiven less restrictive generic conditions on its prolonged system. In this\nsetting we find that the solution stratifies the manifold according to the\nstrict signature (pointwise) of the solution and does this in way that locally\ngeneralizes the stratification of a model, where the model is, in each case, a\ncorresponding Lie group orbit decomposition of the sphere. Thus the solutions\ngive curved generalizations of such embedded orbit structures. We describe the\nsmooth nature of the strata and determine the geometries of each of the\ndifferent strata types; this includes a metric on the open strata that becomes\nsingular at the strata boundary, with the latter a type of projective infinity\nfor the given metric. The approach reveals and exploits interesting highly\nnon-linear relationships between different linear geometric partial\ndifferential equations. Apart from their direct significance, the results show\nthat, for the metrizability equation, strong results arising for so-called\nnormal BGG solutions, and the corresponding projective holonomy reduction,\nextend to a far wider class of solutions. The work also provides new results\nfor the projective compactification of scalar-flat metrics.\n