A compactness theorem for locally homogeneous spaces

TL;DR

This paper establishes a compactness theorem for locally homogeneous spaces with bounded sectional curvature, proving existence, uniqueness, and compactness of geometric models in a specific topology.

Contribution

It introduces a compactness theorem for geometric models of locally homogeneous spaces with curvature bounds, including existence and uniqueness results.

Findings

Existence and uniqueness of geometric models for locally homogeneous spaces.

The set of models is compact in the pointed ^{1,\u03b1} topology.

Models are characterized under sectional curvature bounds ||  1.

Abstract

We prove the existence and uniqueness of geometric models of local isometry classes of locally homogeneous spaces with sectional curvature $|\operatorname{sec}|\leq 1$. Moreover, we show that the set of geometric models is compact in the pointed $\mathcal{C}^{1,\alpha}$-topology.