Einstein metrics in projective geometry

TL;DR

This paper establishes a connection between normal solutions of a projectively invariant differential equation and Einstein metrics within the same projective class, enriching the understanding of Einstein metrics in projective geometry.

Contribution

It proves that non-degenerate normal solutions of a specific BGG equation correspond to Einstein metrics in the projective class, linking differential equations to geometric structures.

Findings

Normal solutions are equivalent to Einstein metrics in the projective class

Connects Einstein condition to projective extensions

Provides a new perspective on Einstein metrics via BGG equations

Abstract

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.