Detecting Einstein geodesics: Einstein metrics in projective and conformal geometry

TL;DR

This paper investigates conditions under which a torsion-free connection's unparametrised geodesics match those of an Einstein metric, introducing projective invariants that serve as necessary and sometimes sufficient criteria.

Contribution

It introduces projective invariants that determine when geodesics correspond to Einstein metrics and explores their relation to conformal geometry and metrisability.

Findings

Identified projective invariants necessary for Einstein metric existence.

Established conditions where these invariants are also sufficient.

Linked projective and conformal geometry in the context of Einstein metrics.

Abstract

Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that the vanishing of these is necessary for the existence of such a metric, and in generic settings the vanishing of these is also sufficient. We also obtain results for the problem of metrisability (without the Einstein condition): We show that the odd Chern type invariants of an affine connection are projective invariants that obstruct the existence of a projectively related Levi-Civita connection. In addition we discuss a concrete link between projective and conformal geometry and the application of this to the projective-Einstein problem.