Einstein connections and involutions via parabolic geometries

Abstract

This paper demonstrates a way of finding Einstein connections (affine connections whose Ricci tensor is non-degenerate and covariantly constant) via parabolic geometry. Extending the results in the projective and conformal cases, it demonstrates that the existence of a preserved involution σ on an associated bundle A leads–under mild conditions–to the construction of a specific Einstein connection, among the Weyl connections of the parabolic geometry. The conditions necessary for the existence of such ‘Einstein involutions’ are then presented, corresponding to a small family of holonomy reductions for the Cartan connection defining the parabolic geometry. This is illustrated with a few examples.