2014/06/30 by Andreas Čap, Andreas Cap, A. Rod Gover
Mathematics · #Affine connection #Boundary (topology) #Compact space #Conformal geometry #Conformal map #Differential geometry #Geodesic #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hypersurface #Nonlinear Partial Differential Equations #Projective differential geometry #math.DG #msc:35N10 #msc:53A20 #msc:53A30 #msc:53B10 #msc:53B21 #msc:58J60
paper · pdf · doi:10.1007/s00208-016-1370-9
published as Math. Ann. 366, 3-4 (2016) 1587-1620 · Substantially revised, including simpler arguments for many of the main results. 32 pages, comments are welcome
arxiv created 2015/03/19 · openalex publication_date 2016/01/29 · openalex created_date 2016/06/24 · arxiv updated 2016/11/08 · openalex updated_date 2026/08/05
Let M be a smooth manifold with boundary ∂ M and interior M. Consider an affine connection ∇ on M for which the boundary is at infinity. Then ∇ is projectively compact of order α if the projective structure defined by ∇ smoothly extends to all of M in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on M. We study the relation between interior geometry and the possibilities for compactification, and then develop the tools that describe the induced geometry on the boundary. We prove that a pseudo--Riemannian metric on M 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 M, which induces a conformal class of (possibly degenerate) bundle metrics on the tangent bundle to the hypersurface ∂ M. Using the asymptotic form, we prove that in the case of metrics, which are projectively compact of order two, this boundary structure is always non--degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non--degenerate boundary geometry gives rise to a (conformal) standard tractor bundle endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.