Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories

TL;DR

This paper introduces a generalized Einstein-Hilbert-Palatini functional for manifolds based on arbitrary graded algebras, revealing topological and geometric obstructions to certain Einstein manifold structures.

Contribution

It generalizes the EHP functional to A-valued cases and establishes conditions under which it is non-zero, linking algebraic properties to geometric obstructions.

Findings

Non-null A-EHP functional implies dimension constraints for weak (k,s)-solvable algebras.

Classical geometries mostly satisfy conditions for non-null A-EHP, except certain semi-Riemannian cases.

Torsionless A-EHP is non-null only on Kähler manifolds of dimension 2 or 4.

Abstract

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.