Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories

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 K\"{a}hler of dimension $n=2,4$. We present our results as obstructions to $M$ being an Einstein manifold relative to geometries other than semi-Riemannian.