Projective Compactifications and Einstein metrics

Abstract

For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e.\ geodesic path data) of the given connection. The definition of projective compactness involves a real parameter $\alpha$ called the order of projective compactness. For volume preserving connections, this order is captured by a notion of volume asymptotics that we define. These ideas apply to complete pseudo-Riemannian spaces, via the Levi-Civita connection, and thus provide a notion of compactification alternative to conformal compactification. For each order $\alpha$, we provide an asymptotic form of a metric which is sufficient for projective compactness of the given order, thus also providing many local examples. Distinguished classes of projectively compactified geometries of orders one and two are associated with Ricci-flat connections and non--Ricci--flat Einstein metrics, respectively. Conversely, these geometric conditions are shown to force the indicated order of projective compactness. These special compactifications are shown to correspond to normal solutions of classes of natural linear PDE (so-called first BGG equations), or equivalently holonomy reductions of projective Cartan/tractor connections. This enables the application of tools already available to reveal considerable information about the geometry of the boundary at infinity. Finally, we show that metrics admitting such special compactifications always have an asymptotic form as mentioned above.

30 Pages. v2: Added a necessary technical assumption in Theorem 2.6. Adapted numbering to version to be published in J. reine angew. math. (Crelle's J.)

References in corpus (4)

• The principle of equivalence and projective structure in space-times
• Einstein metrics in projective geometry
• Metrisability of three-dimensional projective structures
• Holonomy reductions of Cartan geometries and curved orbit decompositions

Cited by in corpus (18)

• C-projective geometry
• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Metric projective geometry, BGG detour complexes and partially massless gauge theories
• Nearly Kähler Geometry and $(2,3,5)$-Distributions via Projective Holonomy
• Projective Compactness and Conformal Boundaries
• C-Projective Compactification; (quasi--)Kaehler Metrics and CR boundaries
• Asymptotically simple spacetimes and mass loss due to gravitational waves
• Geometric Theory of Weyl Structures
• Scalar Curvature and Projective Compactness
• Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification
• Asymptotic symmetries of projectively compact order one Einstein manifolds
• Peeling property and asymptotic symmetries with a cosmological constant
• Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
• Boundary rigidity, and non-rigidity, of projective structures
• A relative mass cocycle and the mass of asymptotically hyperbolic manifolds
• Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity
• Parabolic Compactification of Homogeneous Spaces
• Tractor calculus, BGG complexes, and the cohomology of Kleinian groups