Bianchi universe
1. History
The Bianchi universes are based on the Bianchi classification of $3$-manifolds (or more generally any manifold) into homogeneous ones, which was done by Luigi Bianchi in 1898 in his paper Sugli spazii a tre dimensioni che ammettono un gruppo continuo di movimenti (On three-dimensional spaces which admit a continuous group of motions). This idea was later extended to spacetimes
2. Topology
Bianchi universes have the topology $\mathbb R \times G$, with $G$ a $3$-dimensional Lie group.
3. Metrics and coordinates
The metric of Bianchi universes is of the form
$$ds^2 = -dt^2 + h_t$$with $h_t$ a family of left-invariant Riemannian metric on the group $G$, parametrized by $t$.
4. Tensor quantities
5. Symmetries
All Bianchi universes are homogeneous, with the usual translation Killing vectors for their appropriate topology (for instance $\partial_x$, $\partial_y$, $\partial_z$