2007/05/16 by Stuart Armstrong, Armstrong, Stuart · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #math.AP #math.DG #msc:51F25 #msc:51F99 #msc:51M15 #msc:53B05 #msc:53B10 #msc:53B15 #msc:53B35
paper · pdf · doi:10.48550/arxiv.0705.2390
Newest version, with SO*(2m) included
arxiv created 2008/08/14 · arxiv updated 2009/12/01
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 \mcA gives rise, given certain algebraic conditions, to a unique preferred affine connection ∇ with covariantly constant rho-tensor P, compatible with the algebraic bracket on \mcA. 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 \mfg is \mfsl(m,\mbbF), preserved complex structures anti-commuting with the skew-form for \mfg=\mfsp(2m,\mbbF), 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.