Model geometry

Definition

A model geometry $(G,X)$ is a manifold $X$ together with a Lie group $G$ of diffeomorphisms of $X$ such that:

1. $X$ is connected and simply connected
2. $G$ acts transitively on $X$, with compact point stabilizers
3. $G$ is not contained in any larger group of diffeomorphisms of $X$ with compact point-stabilizers
4. There exists at least one compact manifold modeled on $(G,X)$