|Title||Metric projective geometries and projective compactification|
|Presenter||Keegan Flood (The University of Auckland)|
|Author(s)||Mr Keegan Flood|
Two affine connections on a manifold are projectively equivalent if they have the same geodesics up to reparametrization. A projective manifold is a smooth manifold equipped with a class of projectively equivalent torsion-free affine connections. Given a projective manifold it is natural to ask whether there is a connection in its projective class arising as the Levi-Civita connection of a metric. The answer is related to the existence of a solution to a geometric PDE known as the metrizability equation. We show that under certain assumptions the degeneracy locus of a solution to the metrizability equation is a smoothly embedded separating hypersurface endowed with either a projective or a conformal structure. In doing so we relate scalar curvature conditions on the interior of a manifold with boundary to the manner in which the projective structure of the interior extends to the boundary. This is joint work with A. R. Gover.