Convergence of locally homogeneous spaces

Abstract

We study three different topologies on the moduli space ${\mathcal{H}}_m^{\mathrm{loc}}$ of equivariant isometry classes of $m$-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed ${\mathcal{C}}^{k,\alpha }$-topology, for some $k>1$, which do not admit any convergent subsequence in the pointed ${\mathcal{C}}^{k+1}$-topology.

17 pages. Some minor modifications. Accepted for publication in Geom. Dedicata