Convergence of locally homogeneous spaces

TL;DR

This paper explores different topologies on the moduli space of locally homogeneous Riemannian spaces and presents novel examples of convergence behavior in certain topologies.

Contribution

It introduces new examples of locally homogeneous spaces converging in the pointed ^{k,lpha} topology without convergence in the ^{k+1} topology.

Findings

First examples of convergence in ^{k,lpha} topology without ^{k+1} convergence.

Analysis of three topologies on the moduli space of locally homogeneous spaces.

Insights into the convergence properties of locally homogeneous Riemannian spaces.

Abstract

We study three different topologies on the moduli space $\mathscr{H}^{\rm loc}_m$ 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.