Generalised Einstein condition and cone construction for parabolic geometries

Abstract

This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution $\sigma$ of the adjoint bundle $\mc{A}$ gives rise, given certain algebraic conditions, to a unique preferred affine connection $\nabla$ with covariantly constant rho-tensor $\mathsf{P}$, compatible with the algebraic bracket on $\mc{A}$. These conditions can reasonably be considered the generalisations of the Einstein condition, and recreate the standard Einstein condition in conformal geometry. The existence of such an involution is implies by some simpler structures: preserved metrics when the overall algebra $\mf{g}$ is $\mf{sl}(m,\mbb{F})$, preserved complex structures anti-commuting with the skew-form for $\mf{g}=\mf{sp}(2m,\mbb{F})$, and preserved subundles of the tangent bundle, of a certain rank, for all the other non-exceptional simple Lie algebras. Examples of Einstein involutions are constructed or referenced for several geometries. The existence of cone constructions for certain Einstein involutions is then demonstrated.