Bianchi Classification and Homogeneous Geometry

Bianchi classification organizes spatially homogeneous cosmologies by the Lie algebra acting transitively on each three-dimensional spatial slice. The structure constants determine the allowed invariant one-forms and spatial-curvature terms; the time-dependent metric coefficients then obey ordinary differential equations. The classification concerns the local symmetry algebra, not the global topology of a spatial quotient.

Required background. Einstein’s Equation supplies the reduced evolution and constraints; Continuum Kinematics and Balance Laws supplies homogeneous matter variables.

Helpful background. The Riemann Curvature Tensor supplies intrinsic spatial curvature; Exact Solutions of Einstein’s Equations supplies the family comparison; Killing Fields and Isometry Groups supplies transitive group actions.

Invariant frames and Maurer–Cartan signs

Let $e_a$ be a basis of invariant vector fields for a simply transitive three-dimensional group action and $\omega^a$ its dual coframe. Fix the convention

$$[e_b,e_c]=C^a{}_{bc}e_a, \qquad d\omega^a=-\frac12C^a{}_{bc}\, \omega^b\wedge\omega^c.$$

The minus sign follows by evaluating $d\omega^a(e_b,e_c)=-\omega^a([e_b,e_c])$. Declaring it is essential: changing from left- to right-invariant fields reverses the structure constants.

A homogeneous metric can be written

$$ds^2=-N^2(t)dt^2+h_{ab}(t) \big(\omega^a+N^a dt\big) \big(\omega^b+N^b dt\big).$$

In a comoving orthogonal gauge one sets $N^a=0$, but tilted fluids or other foliations need not permit that simplification. Homogeneity removes spatial dependence from $h_{ab}$ in this invariant basis; it does not require the metric to be diagonal.

Structure-constant decomposition

After choosing an orientation and a positive basis metric for the algebraic decomposition, write

$$C^a{}_{bc}=\epsilon_{bcd}n^{da} +a_b\delta^a_c-a_c\delta^a_b,$$

with $n^{ab}=n^{ba}$. The Jacobi identity is equivalent to

$$n^{ab}a_b=0.$$

Class A algebras have $a_b=0$; equivalently $C^a{}_{ab}=0$, so they are unimodular. In a canonical diagonal basis, the class A types can be represented by the signs of $(n_1,n_2,n_3)$:

• I: $(0,0,0)$;
• II: $(1,0,0)$;
• VI$_0$: $(1,-1,0)$;
• VII$_0$: $(1,1,0)$;
• VIII: $(1,1,-1)$; and
• IX: $(1,1,1)$.

Permutations and simultaneous nonzero rescalings do not change the isomorphism type. Class B has $a_b\neq0$ and includes types IV, V, VI$_h$, and VII$_h$. Definitions of the parameter $h$ differ by signs or reciprocals between canonical bases, so a type VI$_h$ formula must include its structure constants rather than only the label. The decomposition and cosmological interpretation are established in Ellis and MacCallum 1969, §§2–4, pp. 108–121.

Geometry encoded by the algebra

The structure constants enter the spatial Levi-Civita connection algebraically. Together with $h_{ab}(t)$ they determine ${}^{(3)}R_{ab}$, while the extrinsic curvature carries time derivatives. Einstein’s equations therefore become a Hamiltonian constraint, three momentum constraints, and evolution ODEs. Class A models often admit especially clean Hamiltonian variables; applying a class A reduced action unchanged to class B can lose a boundary or constraint term.

Type I has flat group orbits when the invariant metric is spatially constant in Cartesian form. Type IX uses the $SU(2)$ algebra and admits positively curved isotropic data. Type V admits an open isotropic specialization. An FLRW geometry has a six-dimensional spatial isometry group, so a chosen simply transitive three-dimensional subgroup need not be unique; a Bianchi label describes the selected homogeneous action, not every symmetry of the isotropic limit.

Global topology is additional data. The same local type may or may not admit compact quotients, and quotienting can remove continuous isometries or change geodesic and causal properties without changing the local structure constants.

Independent checks

First verify $d^2\omega^a=0$; this reproduces the Jacobi identity and catches sign or index errors. Next compute $C^a{}_{ab}$ to distinguish class A from B. Finally take an isotropic limit $h_{ab}=a^2\delta_{ab}$ only when that algebra admits it and compare the resulting ${}^{(3)}R$ with $6k/a^2$. A diagonal ansatz should never be assumed merely because $n^{ab}$ has been diagonalized: matter tilt and off-diagonal shear can remain dynamical.

Exercise

For Bianchi II, choose $d\omega^1=-\omega^2\wedge\omega^3$ and $d\omega^2=d\omega^3=0$. Read off the nonzero structure constants and verify $d^2=0$.

Solution

Comparison with $d\omega^a=-\tfrac12C^a{}_{bc}\omega^b\wedge\omega^c$ gives $C^1{}_{23}=1$ and $C^1{}_{32}=-1$. Applying $d$ to $d\omega^1$ gives $-d\omega^2\wedge\omega^3+\omega^2\wedge d\omega^3=0$; the other two equations are immediate. Hence the Jacobi identity holds.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models.” In Theoretical and Observational Cosmology, edited by Marc Lachièze-Rey, 1–116. Dordrecht: Springer, 1999. DOI; Open PDF.
• Ellis, George F. R., and Malcolm A. H. MacCallum. “A Class of Homogeneous Cosmological Models.” Communications in Mathematical Physics 12 (1969): 108–141. DOI.