The eight Thurston model geometries #
TauCeti.ModelGeometry is the closed list of spherical, Euclidean, hyperbolic,
S² × ℝ, ℍ² × ℝ, SL₂ℝ~, Nil, and Sol geometries. Its Space and model accessors select
concrete manifolds with the standard analytic Riemannian metrics. The three-dimensional model
vector spaces, analytic manifold structures, and transitive actions of the full Riemannian
isometry groups are available uniformly for an arbitrary member of the list.
The metrics and their homogeneity proofs come from the individual model files. In particular,
the product geometries carry the product metrics, and Nil, Sol, and SL₂ℝ~ keep their distinct
metrics on distinct type synonyms of ℝ³. The acting group is TauCeti.Isom G.model G.Space,
which contains every smooth self-diffeomorphism preserving the metric, rather than just the
transitive subgroups exhibited by those proofs.
This finite index type allows geometric-structure statements to quantify over the eight models
without allowing arbitrary homogeneous spaces. A manifold modeled on G can use
ChartedSpace G.Space M and the existing isometry-action structure groupoid; completeness of
such a structure is an additional condition on its induced metric.
References #
- P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487, Sections 4 and 5.
- W. P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, Princeton University Press (1997), Section 3.8.
The eight Thurston geometries, indexing their standard homogeneous Riemannian models.
- spherical : ModelGeometry
The round three-sphere.
- euclidean : ModelGeometry
Euclidean three-space.
- hyperbolic : ModelGeometry
Hyperbolic three-space.
- sphereProd : ModelGeometry
The product of the round two-sphere and the real line.
- hyperbolicProd : ModelGeometry
The product of the hyperbolic plane and the real line.
- sl2Tilde : ModelGeometry
The universal cover of
SL₂ℝ, with its standard geometry. - nil : ModelGeometry
The real Heisenberg group with its standard left-invariant metric.
- sol : ModelGeometry
The solvable Lie group Sol with its standard left-invariant metric.
Instances For
The concrete manifold underlying a model geometry.
The spherical factors are unit spheres, and the hyperbolic factors use the upper half-space
model. The SL₂ℝ~, Nil, and Sol cases use their metric-bearing type synonyms.
Instances For
The vector space for the charts of a model geometry. It has real dimension three in every case, including the product geometries.
Instances For
The topological chart model. The product cases use Mathlib's ModelProd tag to retain
its canonical product atlas.
Instances For
The boundaryless model with corners used by the standard charts. Products use the product model, so their tangent metrics are the existing product Riemannian metrics.
Instances For
Every model has points, so homogeneity is a nonempty transitive action.
Every model is Hausdorff.
Every model is locally compact, being locally homeomorphic to a finite-dimensional space.
Each model is locally path connected in its standard manifold topology.
All eight models are analytic manifolds in their standard charts.
The standard metric on each model, including the round and product metrics.
The standard metrics of all eight models are analytic.
Each model is homogeneous under its full Riemannian isometry group.
Every model geometry has dimension three.
There are exactly eight model geometries.