5-dimensional geometries III: the fibered geometries

TL;DR

This paper classifies 5-dimensional homogeneous geometries with reducible isotropy representations, revealing new product geometries, infinite families, and distinct maximal geometries within Thurston's framework.

Contribution

It provides a comprehensive classification of fibered 5-dimensional geometries with reducible isotropy, expanding the understanding of Thurston's geometries in higher dimensions.

Findings

Identified a countably infinite family of geometries diffeomorphic to S^3 x S^2.

Discovered an uncountable family of geometries with only a subfamily admitting compact quotients.

Analyzed the non-maximal geometry SO(4)/SO(2) and its realization by two maximal geometries.

Abstract

We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequivalent geometries diffeomorphic to $S^3 \times S^2$; an uncountable family in which only a countable subfamily admits compact quotients; and the non-maximal geometry SO(4)/SO(2) realized by two distinct maximal geometries.