Projective Infinities and b-Calculus

Abstract

For a manifold $\overline{M}$ with boundary $\partial M$ and interior $M$, we introduce and study a weakening of the concept of projective compactness for torsion-free linear connections on $M$, which we call projective pre-compactness. Via the Levi-Civita connection, this concept applies to pseudo-Riemannian metrics on $M$. This is motivated by scattering theory and general relativity (GR), via asymptotic forms of metrics used in these areas. In the general setting of a projectively pre-compact connection $\nabla$ we show that, assuming weak asymptotic conditions on the Ricci curvature, there is an induced projective structure on the boundary. Under a slightly stronger condition on Ricci, we show that the standard tractor bundle and its normal tractor connection arise naturally on this boundary structure. The key ingredient to this is that $\nabla$ admits a smooth extension to the boundary as a linear connection on the tensor product of Melrose's b-tangent bundle with a density bundle, which then restricts to the boundary tractor bundle. A projectively pre-compact pseudo-Riemannian metric (satisfying the conditions on the Ricci curvature) is then shown to induce a holonomy reduction of the boundary projective structure to an indefinite orthogonal group. This endows the boundary with a decomposition into so-called curved orbits, which are either open or embedded hypersurfaces, representing space-like, time-like and light-like infinities in a GR context. We introduce and study a new asymptotic form for such metrics which is available near any boundary point and relate it to an asymptotic form used in general relativity, which is only available near boundary points in the open curved orbits. We show that, in that region, projective pre-compactness essentially is equivalent to the asymptotic form from GR, and projective compactness is equivalent to vanishing of the mass aspect.