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 ${\mathrm{F}}^4$). These are the manifolds ${\mathrm{Sol}}_0^4$, ${\mathrm{Sol}}_1^4$, ${\mathrm{Sol}}_{m,n}^4$ 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).