Projective compactness and conformal boundaries

Abstract

For complete affine manifolds we treat a recently introduced general notion of compactification based on the projective differential geometry (i.e. geodesic path data) of the given connection. This involves a real parameter called the order of projective compactness; for volume preserving connections, this order is a measure of asymptotic volume growth. Via the Levi-Civita connection, this concept applies to complete pseudo-Riemannian metrics. So projective compactness provides a geometrically motivated alternative to conformal compactification. We study the fundamental links between interior geometry and the embedding of the boundary hypersurface at infinity, and develop the tools that describe the induced geometry on this boundary. We prove that a pseudo-Riemannian metric which is projectively compact of order two admits a certain asymptotic form. This form was known to be sufficient for projective compactness, so the result establishes that it provides an equivalent characterization. From a projectively compact connection on M, one obtains a projective structure on the compactified manifold with boundary $\overline{M}$. We show this induces a, possibly degenerate, conformal structure on the boundary hypersurface $\partial M$. We prove that in the case of metrics which are projectively compact of order two this boundary structure is in fact always non-degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non-degenerate conformal boundary geometry gives rise to a conformal standard tractor bundle (on the boundary) endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.