Some examples of projective and $c$--projective compactifications of Einstein metrics

TL;DR

This paper constructs various examples of compactifications of Einstein metrics, including non-metric projective compactifications and para--c--projective compactifications, expanding understanding of Einstein metric boundary behaviors.

Contribution

It introduces new types of projective and c--projective compactifications for Einstein metrics, including non-metric and para--c--projective cases, with explicit constructions.

Findings

Eguchi--Hanson instanton admits a non-metric projective compactification

Metric cones over pseudo-Riemannian manifolds admit metric projective compactifications

Constructs para--c--projective compactifications on certain affine bundles

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.