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Thurston geometries in dimension four from a Riemannian perspective

Marie D'haene

Abstract

In this survey we focus on a special class of homogeneous manifolds called Thurston geometries. We give special attention to the four-dimensional Thurston geometries with 4 or 5-dimensional isometry group which are not a product (except for $\mathrm F^4$). These are the manifolds $\mathrm{Sol}^4_0$, $\mathrm{Sol}^4_1$, $\mathrm{Sol}^4_{m,n}$ and $\mathrm{Nil}^4$. We give a description of each of the spaces and we exhibit all Riemannian metrics which are invariant under the action of their associated group (this last part includes some small new results).

Thurston geometries in dimension four from a Riemannian perspective

Abstract

In this survey we focus on a special class of homogeneous manifolds called Thurston geometries. We give special attention to the four-dimensional Thurston geometries with 4 or 5-dimensional isometry group which are not a product (except for ). These are the manifolds , , and . We give a description of each of the spaces and we exhibit all Riemannian metrics which are invariant under the action of their associated group (this last part includes some small new results).
Paper Structure (3 sections, 19 equations, 3 tables)

This paper contains 3 sections, 19 equations, 3 tables.

Table of Contents

  1. Thurston geometries
  2. A Riemannian description of some Thurston geometries
  3. Future research on 4-dimensional Thurston geometries

Theorems & Definitions (1)

  • Remark 1.1