  Consider a manifold with boundary, and such that the interior is equipped
with a pseudo-Riemannian metric. We prove that, under mild asymptotic
non-vanishing conditions on the scalar curvature, if the Levi-Civita connection
of the interior does not extend to the boundary (because for example the
interior is complete) whereas its projective structure does, then the metric is
projectively compact of order 2; this order is a measure of volume growth
toward infinity. The result implies a host of results including that the metric
satisfies asymptotic Einstein conditions, and induces a canonical conformal
structure on the boundary. Underpinning this work is a new interpretation of
scalar curvature in terms of projective geometry. This enables us to show that
if the projective structure of a metric extends to the boundary then its scalar
curvature also naturally and smoothly extends.
