Conjecture on compact locally homogeneous simple pseudo-Riemannian Einstein manifolds

Let $G$ be a simple non-compact Lie group, and let $G/H$ be a homogeneous space. A standard triple $(G,H,L)$ consists of reductive subgroups $H,L\subset G$ with $G=HL$ and $H\cap L$ compact; a standard Clifford–Klein form is obtained from such a triple and a cocompact lattice in $L$. Locally homogeneous compact pseudo-Riemannian Einstein manifolds considered here are quotients $\Gamma\backslash G/H$.

The conjecture. The only compact locally homogeneous simple pseudo-Riemannian Einstein manifolds are of the form $\Gamma\backslash G/H$, determined by either some standard triple $(G,H,L)$ and a cocompact lattice $\Gamma$ in $L$ or a deformation of such $\Gamma$, or by $(G,\hat H,\hat L)$, where $\hat H$ and $\hat L$ differ from $H$ and $L$ by a compact factor.

The paper proves that every standard triple of a simple non-compact Lie group determines a compact Clifford–Klein form with at least one Einstein metric. The proposed classification is presented as a hypothesis guided by Kobayashi's conjecture, and its general validity remains open.