Generalised Einstein condition and cone construction for parabolic geometries [article]

Stuart Armstrong
2008 arXiv   pre-print
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 σ of the adjoint bundle A gives rise, given certain algebraic conditions, to a unique preferred affine connection ∇ with covariantly constant rho-tensor P, compatible with the algebraic bracket on A. These conditions can reasonably be considered the generalisations of the Einstein condition, and recreate the
more » ... tandard Einstein condition in conformal geometry. The existence of such an involution is implies by some simpler structures: preserved metrics when the overall algebra g is sl(m,F), preserved complex structures anti-commuting with the skew-form for g=sp(2m,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.
arXiv:0705.2390v5 fatcat:nrtfhjz4avd67mxzrekqj66kb4