Bianchi classification

Figure 1. The parameter space as a 3-plane (class A) and an orthogonal half 3-plane (class B) in R⁴ with coordinates (n⁽¹⁾, n⁽²⁾, n⁽³⁾, a), showing the canonical representatives of each Bianchi type.

The three-dimensional Bianchi spaces each admit a set of three Killing vector fields ${\displaystyle \xi _{i}^{(a)}}$ which obey the following property:

${\displaystyle \left({\frac {\partial \xi _{i}^{(c)}}{\partial x^{k}}}-{\frac {\partial \xi _{k}^{(c)}}{\partial x^{i}}}\right)\xi _{(a)}^{i}\xi _{(b)}^{k}=C_{\ ab}^{c}}$

where ${\displaystyle C_{\ ab}^{c}}$, the "structure constants" of the group, form a constant order-three tensor antisymmetric in its lower two indices. For any three-dimensional Bianchi space, ${\displaystyle C_{\ ab}^{c}}$ is given by the relationship

${\displaystyle C_{\ ab}^{c}=\varepsilon _{abd}n^{cd}-\delta _{a}^{c}a_{b}+\delta _{b}^{c}a_{a}}$

where ${\displaystyle \varepsilon _{abd}}$ is the Levi-Civita symbol, ${\displaystyle \delta _{a}^{c}}$ is the Kronecker delta, and the vector ${\displaystyle a_{a}=(a,0,0)}$ and diagonal tensor ${\displaystyle n^{cd}}$ are described by the following table, where ${\displaystyle n^{(i)}}$ gives the ith eigenvalue of ${\displaystyle n^{cd}}$;[1] the parameter a runs over all positive real numbers:

Bianchi type	${\displaystyle a}$	${\displaystyle n^{(1)}}$	${\displaystyle n^{(2)}}$	${\displaystyle n^{(3)}}$	notes
I	0	0	0	0	describes Euclidean space
II	0	1	0	0
III	1	0	1	-1	the subcase of type VIa with ${\displaystyle a=1}$
IV	1	0	0	1
V	1	0	0	0	has a hyper-pseudosphere as a special case
VI0	0	1	-1	0
VIa	${\displaystyle a}$	0	1	-1	when ${\displaystyle a=1}$, equivalent to type III
VII0	0	1	1	0	has Euclidean space as a special case
VIIa	${\displaystyle a}$	0	1	1	has a hyper-pseudosphere as a special case
VIII	0	1	1	-1
IX	0	1	1	1	has 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 ${\displaystyle C_{ab}^{c}=-C_{ba}^{c}}$, there are nine independent constants ${\displaystyle C_{ab}^{c}}$. These can be equivalently represented by the nine components of an arbitrary constant matrix C^ab:

   ${\displaystyle C_{ab}^{c}=\varepsilon _{abd}C^{dc},}$

   where ε_abd is the totally antisymmetric three-dimensional Levi-Civita symbol (ε₁₂₃ = 1). Substitution of this expression for ${\displaystyle C_{ab}^{c}}$ into the Jacobi identity, results in

   ${\displaystyle \varepsilon _{abd}C^{bd}C^{ac}=0.}$

2. The structure constants can be transformed as:

   ${\displaystyle C^{ab}=\left(\det {A}\right)^{-1}A_{m}^{a}A_{n}^{b}{\acute {C}}^{mn}.}$

   Appearance of det A in this formula is due to the fact that the symbol ε_abd transforms as tensor density: ${\displaystyle \varepsilon _{abc}=\left(\det {A}\right)D_{a}^{m}D_{b}^{n}D_{c}^{d}{\acute {\varepsilon }}_{mnd}}$, where έ_mnd ≡ ε_mnd. By this transformation it is always possible to reduce the matrix C^ab to the form:

   ${\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&C^{22}&C^{23}\\0&C^{32}&C^{33}\end{bmatrix}}.}$

   After such a choice, one still have the freedom of making triad transformations but with the restrictions ${\displaystyle A_{2}^{1}=A_{3}^{1}=0}$ and ${\displaystyle A_{1}^{2}=A_{1}^{3}=0.}$
3. Now, the Jacobi identities give only one constraint:

   ${\displaystyle \left(C^{23}-C^{32}\right)n_{1}=0.}$

4. If n₁ ≠ 0 then C²³ – C³² = 0 and by the remaining transformations with ${\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0,\quad {\bar {a}},{\bar {b}}={\bar {2}},{\bar {3}}}$, the 2 × 2 matrix ${\displaystyle C^{{\bar {a}}{\bar {b}}}}$ in C^ab can be made diagonal. Then

   ${\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&n_{2}&0\\0&0&n_{3}\end{bmatrix}}.}$

   The diagonality condition for C^ab is preserved under the transformations with diagonal ${\displaystyle A_{b}^{a}}$. Under these transformations, the three parameters n₁, n₂, n₃ change in the following way:

   ${\displaystyle n_{a}=\left(A_{1}^{1}A_{2}^{2}A_{3}^{3}\right)\left(A_{a}^{a}\right)^{2}{\acute {n}}_{a},{\text{no summation over}}\ a.}$

   By these diagonal transformations, the modulus of any n_a (if it is not zero) can be made equal to unity. Taking into account that the simultaneous change of sign of all n_a produce nothing new, one arrives to the following invariantly different sets for the numbers n₁, n₂, n₃ (invariantly different in the sense that there is no way to pass from one to another by some transformation of the triad ${\displaystyle e_{a}^{\bar {a}}=A_{\bar {b}}^{\bar {a}}e_{a}^{\bar {b}}}$), that is to the following different types of homogeneous spaces with diagonal matrix C^ab:

   ${\displaystyle {\begin{matrix}Bianchi\ IX&:&(n_{1},n_{2},n_{3})&=&(1,1,1),\\Bianchi\ VIII&:&(n_{1},n_{2},n_{3})&=&(1,1,-1),\\Bianchi\ VII_{0}&:&(n_{1},n_{2},n_{3})&=&(1,1,0),\\Bianchi\ VI_{0}&:&(n_{1},n_{2},n_{3})&=&(1,-1,0),\\Bianchi\ II&:&(n_{1},n_{2},n_{3})&=&(1,0,0).\end{matrix}}}$

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 (0, 1, 1), (0, 1, −1), (0, 0, 1) and (0, 0, 0). The first three repeat the types VII₀, VI₀, II. Consequently, only one new type arises:

   ${\displaystyle Bianchi\ I\ :\ (n_{1},n_{2},n_{3})\ =\ (0,0,0).}$

6. The only case left is n₁ = 0 and C²³ – C³² ≠ 0. Now the 2 × 2 matrix ${\displaystyle C^{{\bar {a}}{\bar {b}}}({\bar {a}},{\bar {b}}=2,3)}$ is non-symmetric and it cannot be made diagonal by transformations using ${\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0}$. However, its symmetric part can be diagonalized, that is the 3 × 3 matrix C^ab can be reduced to the form:

   ${\displaystyle C^{ab}={\begin{bmatrix}0&0&0\\0&n_{2}&a\\0&-a&n_{3}\end{bmatrix}},}$

   where a is an arbitrary number. After this is done, there still remains the possibility to perform transformations with diagonal ${\displaystyle A_{\bar {b}}^{\bar {a}}}$, under which the quantities n₂, n₃ and a change as follows:

   ${\displaystyle n_{2}=\left(A_{1}^{1}A_{2}^{2}A_{3}^{3}\right)^{-1}\left(A_{2}^{2}\right)^{2}{\acute {n}}_{2},\quad n_{3}=\left(A_{1}^{1}A_{2}^{2}A_{3}^{3}\right)^{-1}\left(A_{3}^{3}\right)^{2}{\acute {n}}_{3},\quad a=\left(A_{1}^{1}\right)^{-1}{\acute {a}}.}$

   These formulas show that for nonzero n₂, n₃, a, the combination a²(n₂n₃)⁻¹ is an invariant quantity. By a choice of ${\displaystyle A_{1}^{1}}$, one can impose the condition a > 0 and after this is done, the choice of the sign of ${\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}$ permits one to change both signs of n₂ and n₃ simultaneously, that is the set (n₂ , n₃) is equivalent to the set (−n₂,−n₃). It follows that there are the following four different possibilities:

   ${\displaystyle (a,n_{2},n_{3})=(a,0,0),(a,0,1),(a,1,1),(a,1,-1).}$

   For the first two, the number a can be transformed to unity by a choice of the parameters ${\displaystyle A_{1}^{1}}$ and ${\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}$. 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:

   ${\displaystyle {\begin{matrix}Bianchi\ V&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(1,0,0),\\Bianchi\ IV&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(1,0,1),\\Bianchi\ VII&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(a,1,1),\\Bianchi\ III&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(1,1,-1),\\Bianchi\ VI&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(a,1,-1).\end{matrix}}}$

   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.

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:

${\displaystyle ds^{2}=\gamma _{ab}\xi _{i}^{(a)}\xi _{k}^{(b)}dx^{i}dx^{k}}$

(where ${\displaystyle \xi _{i}^{(a)}dx^{i}}$　are 1-forms), the Ricci curvature tensor ${\displaystyle R_{ik}}$ is given by:

${\displaystyle R_{ik}=R_{(a)(b)}\xi _{i}^{(a)}\xi _{k}^{(b)}}$

${\displaystyle R_{(a)(b)}={\frac {1}{2}}\left[C_{\ \ b}^{cd}\left(C_{cda}+C_{dca}\right)+C_{\ cd}^{c}\left(C_{ab}^{\ \ d}+C_{ba}^{\ \ d}\right)-{\frac {1}{2}}C_{b}^{\ cd}C_{acd}\right]}$

where the indices on the structure constants are raised and lowered with ${\displaystyle \gamma _{ab}}$ which is not a function of ${\displaystyle x^{i}}$.