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 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