2018/08/31 by Yuri X. Martins, Rodney J. Biezuner · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.DG #math.MP
paper · pdf · doi:10.1016/j.geomphys.2019.04.012
arxiv created 2018/10/24 · arxiv updated 2019/10/09
In this article we introduce A-valued Einstein-Hilbert-Palatini functional (A-EHP) over a n-manifold M, where A is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if A is weak (k,s)-solvable, then A-EHP is non-null only if n<k+s+3. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for k=1 and s=2, including Hitchin's generalized complex geometry, Pantilie's generalized quaternionic geometries and all other generalized Cayley-Dickson geometries. We also prove that if A is concrete in some sense, then a torsionless version of A-EHP is non-null only if M is Kähler of dimension n=2,4. We present our results as obstructions to M being an Einstein manifold relative to geometries other than semi-Riemannian.