Abstract
We construct several examples of compactifications of Einstein metrics. We show that the Eguchi–Hanson instanton admits a projective compactification which is non-metric, and that a metric cone over any (pseudo)-Riemannian manifolds admits a metric projective compactification. We construct a para-c-projective compactification of neutral signature Einstein metrics canonically defined on certain rank-n affine bundles M over n-dimensional manifolds endowed with projective structures.
Highlights
There are several notions of compactifications of a Riemannian manifold (M, g)
In a conformal compactification (M, g), one has that M is a manifold with boundary such that M is the interior of M, and there is a defining function T for the boundary such that the metric g = T 2g smoothly extends to the boundary ∂M of M (T being a defining function for ∂M means that ∂M = Z(T ) := {p ∈ M : T (p) = 0} and dT is nowhere zero on ∂M .) The geodesics of g do not correspond to geodesics of g, but the angles are preserved
The metric g extends to the boundary. This construction leads to global examples of metric projective compactifications
Summary
There are several notions of compactifications of a (pseudo) Riemannian manifold (M, g). In a conformal compactification (M , g), one has that M is a manifold with boundary such that M is the interior of M , and there is a defining function T for the boundary such that the metric g = T 2g smoothly extends to the boundary ∂M of M (T being a defining function for ∂M means that ∂M = Z(T ) := {p ∈ M : T (p) = 0} and dT is nowhere zero on ∂M .) The geodesics of g do not correspond to geodesics of g, but the angles are preserved This kind of compactification has proven to be useful in studying the causal structure of space times in general relativity [21], and quantum field theory [23].
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