5-dimensional geometries II: the non-fibered geometries

Abstract

We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while those with trivial isotropy are simply-connected solvable Lie groups of the form $\mathbb{R}^3 \rtimes \mathbb{R}^2$ or $N \rtimes \mathbb{R}$ where $N$ is nilpotent.