5-dimensional geometries II: the non-fibered geometries

TL;DR

This paper classifies 5-dimensional homogeneous geometries with specific isotropy representations, identifying irreducible symmetric spaces and certain solvable Lie groups as the main types.

Contribution

It provides a detailed classification of 5D geometries with irreducible or trivial isotropy, expanding the understanding of Thurston's geometries in higher dimensions.

Findings

Irreducible isotropy geometries are symmetric spaces.

Trivial isotropy geometries are certain solvable Lie groups.

Complete classification of these geometries in 5D.

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.