58 research outputs found
Einstein metrics in projective geometry
It is well known that pseudo-Riemannian metrics in the projective class of a
given torsion free affine connection can be obtained from (and are equivalent
to) the solutions of a certain overdetermined projectively invariant
differential equation. This equation is a special case of a so-called first BGG
equation. The general theory of such equations singles out a subclass of
so-called normal solutions. We prove that non-degerate normal solutions are
equivalent to pseudo-Riemannian Einstein metrics in the projective class and
observe that this connects to natural projective extensions of the Einstein
condition.Comment: 10 pages. Adapted to published version. In addition corrected a minor
sign erro
Some Examples of Projective and c-projective Compactifications of Einstein Metrics
Funder: University of CambridgeWe 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----projective compactification of neutral signature Einstein metrics
canonically defined on certain rank-- affine bundles over
-dimensional manifolds endowed with projective structures
- …