Bianchi classification


In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras. The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898.
The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of complex Lie algebras.

Classification in dimension less than 3

All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as a semidirect product of R2 and R, with R acting on R2 by some 2 by 2 matrix M. The different types correspond to different types of matrices M, as described below.
The classification of 3-dimensional complex Lie algebras is similar except that types VIII and IX become isomorphic, and types VI and VII both become part of a single family of Lie algebras.
The connected 3-dimensional Lie groups can be classified as follows: they are a quotient of the corresponding simply connected Lie group by a discrete subgroup of the center, so can be read off from the table above.
The groups are related to the 8 geometries of Thurston's geometrization conjecture. More precisely, seven of the 8 geometries can be realized as a left-invariant metric on the simply connected group. The Thurston geometry of type S2×R cannot be realized in this way.

Structure constants

The three-dimensional Bianchi spaces each admit a set of three Killing vector fields which obey the following property:
where, the "structure constants" of the group, form a constant order-three tensor antisymmetric in its lower two indices. For any three-dimensional Bianchi space, is given by the relationship
where is the Levi-Civita symbol, is the Kronecker delta, and the vector and diagonal tensor are described by the following table, where gives the ith eigenvalue of ; the parameter a runs over all positive real numbers:
Bianchi typenotes
I0000describes Euclidean space
II0100
III101-1the subcase of type VIa with
IV1001
V1000has a hyper-pseudosphere as a special case
VI001-10
VIa01-1when, equivalent to type III
VII00110has Euclidean space as a special case
VIIa011has a hyper-pseudosphere as a special case
VIII011-1
IX0111has a hypersphere as a special case

The standard Bianchi classification can be derived from the structural constants in the following six steps:

  1. Due to the antisymmetry, there are nine independent constants. These can be equivalently represented by the nine components of an arbitrary constant matrix Cab:
    where εabd is the totally antisymmetric three-dimensional Levi-Civita symbol. Substitution of this expression for into the Jacobi identity, results in
  2. The structure constants can be transformed as:
    Appearance of det A in this formula is due to the fact that the symbol εabd transforms as tensor density:, where έmnd ≡ εmnd. By this transformation it is always possible to reduce the matrix Cab to the form:
    After such a choice, one still have the freedom of making triad transformations but with the restrictions and
  3. Now, the Jacobi identities give only one constraint:
  4. If n1 ≠ 0 then C23 – C32 = 0 and by the remaining transformations with, the 2 × 2 matrix in Cab can be made diagonal. Then
    The diagonality condition for Cab is preserved under the transformations with diagonal. Under these transformations, the three parameters n1, n2, n3 change in the following way:
    By these diagonal transformations, the modulus of any na can be made equal to unity. Taking into account that the simultaneous change of sign of all na produce nothing new, one arrives to the following invariantly different sets for the numbers n1, n2, n3, that is to the following different types of homogeneous
    spaces with diagonal matrix Cab:
  5. Consider now the case n1 = 0. It can also happen in that case that C23 – C32 = 0. This returns to the situation already analyzed in the previous step but with the additional condition n1 = 0. Now, all essentially different types for the sets n1, n2, n3 are,, and. The first three repeat the types VII0, VI0, II. Consequently, only one new type arises:
  6. The only case left is n1 = 0 and C23 – C32 ≠ 0. Now the 2 × 2 matrix is non-symmetric and it cannot be made diagonal by transformations using. However, its symmetric part can be diagonalized, that is the 3 × 3 matrix Cab can be reduced to the form:
    where a is an arbitrary number. After this is done, there still remains the possibility to perform transformations with diagonal, under which the quantities n2, n3 and a change as follows:
    These formulas show that for nonzero n2, n3, a, the combination a2−1 is an invariant quantity. By a choice of, one can impose the condition a > 0 and after this is done, the choice of the sign of permits one to change both signs of n2 and n3 simultaneously, that is the set is equivalent to the set. It follows that there are the following four different possibilities:
    For the first two, the number a can be transformed to unity by a choice of
    the parameters and. For the second two possibilities, both of these parameters are already fixed and a remains an invariant and arbitrary positive number. Historically these four types of homogeneous spaces have been classified as:
    Type III is just a particular case of type VI corresponding to a = 1. Types VII and VI contain an infinity of invariantly different types of algebras corresponding to the arbitrariness of the continuous parameter a. Type VII0 is a particular case of VII corresponding to a = 0 while type VI0 is a particular case of VI corresponding also to a = 0.

Curvature of Bianchi spaces

The Bianchi spaces have the property that their Ricci tensors can be separated into a product of the basis vectors associated with the space and a coordinate-independent tensor.
For a given metric:
, the Ricci curvature tensor is given by:
where the indices on the structure constants are raised and lowered with which is not a function of.

Cosmological application

In cosmology, this classification is used for a homogeneous spacetime of dimension 3+1. The 3-dimensional Lie group is as the symmetry group of the 3-dimensional spacelike slice, and the Lorentz metric satisfying the Einstein equation is generated by varying the metric components as a function of t. The Friedmann–Lemaître–Robertson–Walker metrics are isotropic, which are particular cases of types I, V, and IX. The Bianchi type I models include the Kasner metric as a special case.
The Bianchi IX cosmologies include the Taub metric. However, the dynamics near the singularity is approximately governed by a series of successive Kasner periods. The complicated dynamics,
which essentially amounts to billiard motion in a portion of hyperbolic space, exhibits chaotic behaviour, and is named Mixmaster; its analysis is referred to as the BKL analysis after Belinskii, Khalatnikov and Lifshitz.
More recent work has established a relation of gravity theories near a spacelike singularity with Lorentzian Kac–Moody algebras, Weyl groups and hyperbolic Coxeter groups.
Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.