Thurston geometries in dimension four from a Riemannian perspective

2024/01/11 by D'haene, Marie
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C42

paper · doi:10.48550/arxiv.2401.05977

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 F⁴). These are the manifolds Sol⁴₀, Sol⁴₁, Sol⁴_m,n and Nil⁴. 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).

Related

• Homogeneous hypersurfaces of the four-dimensional Thurston geometry \rm Sol₀⁴
• Distinguishing 4-dimensional geometries via profinite completions
• Classical Notions and Problems in Thurston Geometries
• How to see the eight Thurston geometries
• 5-dimensional geometries I: the general classification