Pseudo-Riemannian $\mathrm{G}_{2(2)}$-manifolds with dimension at most $21$

Abstract: Let $G_{2(2)}$ be the non-compact connected simple Lie group of type $G_2$ over $\mathbb{R}$, and let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $G_{2(2)}$-action with a dense orbit. For the case $\dim(M) \leq 21$, we provide a full description of the manifold $M$, its geometry and its $G_{2(2)}$-action. The latter are always given in terms of a Lie group geometry related to $G_{2(2)}$, and in one case $M$ is essentially the quotient of $\widetilde{\mathrm{S0}}_0(3,4)$ by a lattice.