Isotropic geometry

All the Thurston geometries are homogenous (i.e. all the points play the same role). In an isotropic geometry $X$, all the directions also play the same role. Said differently, the stabilizer of a point $x \in X$ in the geometry is isomorphic to $O(3)$ acting by isometries on the tangent space $T_xX$.

The euclidean space $\mathbb E^3$

The euclidean space corresponds to the “usual” 3D geometry we learn at school. It is the geometry of the flat real 3-dimensional vector space $\mathbb R^3$.

The spherical space $S^3$

The space $S^3$ is the 3-dimensional analogue of the usual sphere in 2 dimensions. It is an isotropic space (all the directions play the same role). Among the eight Thurston geometries, it is the only compact one. It can be seen as the unit tangent bundle of the 2-sphere. This point of view is also knwon as the Hopf fibration.

The hyperbolic space $\mathbb H^3$

The hyperbolic space $\mathbb H^3$ is the three-dimensional analog of the hyperbolic plane. It is an isotropic space (all the directions play the same role). Among the eight geometries, this is probably the one which has the richest class of lattices.