Differential Geometry, Lie Theory and Low-Dimensional Topology, Melbourne, Australia, 2016

Register for: CADG16

Details of talk

TitleMetric projective geometries and projective compactification
PresenterKeegan 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.