Inmathematics, theBianchi classification provides a list of all real 3-dimensionalLie algebras (up toisomorphism). 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. (Sometimes two of the groups are included in the infinite families, giving 9 instead of 11 classes.) The classification is important in geometry and physics, because the associatedLie groups serve as symmetry groups of 3-dimensionalRiemannian manifolds. It is named forLuigi Bianchi, who worked it out in 1898.

The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications ofcomplex Lie algebras.

• Dimension 0: The only Lie algebra is theabelian Lie algebraR⁰.
• Dimension 1: The only Lie algebra is the abelian Lie algebraR¹, withouter automorphism group the multiplicative group of non-zero real numbers.
• Dimension 2: There are two Lie algebras:
  • (1) The abelian Lie algebraR², with outer automorphism groupGL₂(R).
  • (2) Thesolvable Lie algebra of 2×2 upper triangular matrices of trace 0. It has trivial center and trivial outer automorphism group. Theassociatedsimply connectedLie group is theaffine group of the line.

All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as asemidirect product ofR² andR, withR acting onR² by some 2 by 2 matrixM. The different types correspond to different types of matricesM, as described below.

• Type I: This is the abelian and unimodular Lie algebraR³. The simply connected group has centerR³ and outer automorphism group GL₃(R). This is the case whenM is 0.
• Type II: TheHeisenberg algebra, which isnilpotent and unimodular. The simply connected group has centerR and outer automorphism group GL₂(R). This is the case whenM is nilpotent but not 0 (eigenvalues all 0).
• Type III: This algebra is a product ofR and the 2-dimensional non-abelian Lie algebra. (It is a limiting case of type VI, where one eigenvalue becomes zero.) It issolvable and not unimodular. The simply connected group has centerR and outer automorphism group the group of non-zero real numbers. The matrixM has one zero and one non-zero eigenvalue.
• Type IV: The algebra generated by [y,z] = 0, [x,y] =y, [x,z] =y +z. It is solvable and not unimodular. The simply connected group has trivial center and outer automorphism group the product of the reals and a group of order 2. The matrixM has two equal non-zero eigenvalues, but is notdiagonalizable.
• Type V: [y,z] = 0, [x,y] =y, [x,z] =z. Solvable and not unimodular. (A limiting case of type VI where both eigenvalues are equal.) The simply connected group has trivial center and outer automorphism group the elements of GL₂(R) of determinant +1 or −1. The matrixM has two equal eigenvalues, and is diagonalizable.
• Type VI: An infinite family: semidirect products ofR² byR, where the matrixM has non-zero distinct real eigenvalues with non-zero sum. The algebras are solvable and not unimodular. The simply connected group has trivial center and outer automorphism group a product of the non-zero real numbers and a group of order 2.
• Type VI₀: This Lie algebra is the semidirect product ofR² byR, withR where the matrixM has non-zero distinct real eigenvalues with zero sum. It is solvable and unimodular. It is the Lie algebra of the 2-dimensionalPoincaré group, the group of isometries of 2-dimensionalMinkowski space. The simply connected group has trivial center and outer automorphism group the product of thepositive real numbers with thedihedral group of order 8.
• Type VII: An infinite family: semidirect products ofR² byR, where the matrixM has non-real and non-imaginary eigenvalues. Solvable and not unimodular. The simply connected group has trivial center and outer automorphism group the non-zero reals.
• Type VII₀: Semidirect product ofR² byR, where the matrixM has non-zero imaginary eigenvalues. Solvable and unimodular. This is the Lie algebra of the group of isometries of the plane. The simply connected group has centerZ and outer automorphism group a product of the non-zero real numbers and a group of order 2.
• Type VIII: The Lie algebrasl₂(R) of traceless 2 by 2 matrices, associated to the groupSL₂(R). It issimple and unimodular. The simply connected group is not a matrix group; it is denoted by${\displaystyle \overline{\text{SL}(2,\mathbf{R})}}$, has centerZ and its outer automorphism group has order 2.
• Type IX: The Lie algebra of theorthogonal groupO₃(R). It is denoted by𝖘𝖔(3) and is simple and unimodular. The corresponding simply connected group isSU(2); it has center of order 2 and trivial outer automorphism group, and is aspin group.

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'sgeometrization conjecture. More precisely, seven of the 8 geometries can be realized as a left-invariant metric on the simply connected group (sometimes in more than one way). The Thurston geometry of typeS²×R cannot be realized in this way.

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

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

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

${\displaystyle C_{ab}^c={\epsilon }_{abd}{n}^{cd}-{\delta }_a^c{a}_b+{\delta }_b^c{a}_a}$

where${\displaystyle {\epsilon }_{abd}}$ is theLevi-Civita symbol,${\displaystyle {\delta }_a^c}$ is theKronecker delta, and the vector${\displaystyle a_a=(a,0,0)}$ anddiagonal tensor${\displaystyle n^{cd}}$ are described by the following table, where${\displaystyle n^{(i)}}$ gives theitheigenvalue of${\displaystyle n^{cd}}$;^[1] the parametera runs over all positivereal numbers:

Bianchi type	${\displaystyle a}$	${\displaystyle n^{(1)}}$	${\displaystyle n^{(2)}}$	${\displaystyle n^{(3)}}$	class	notes	graphical (Fig. 1)
I	0	0	0	0	A	describesEuclidean space	at the origin
II	0	1	0	0	A		interval [0,1] along${\displaystyle n^{(1)}}$
III	1	0	1	-1	B	the subcase of type VIa with${\displaystyle a=1}$	projects to fourth quadrant of thea = 0 plane
IV	1	0	0	1	B		vertical open face between first and fourth quadrants of thea = 0 plane
V	1	0	0	0	B	has a hyper-pseudosphere as a special case	the interval (0,1] along the axisa
VI0	0	1	-1	0	A		fourth quadrant of the horizontal plane
VIa	${\displaystyle a}$	0	1	-1	B	when${\displaystyle a=1}$, equivalent to type III	projects to fourth quadrant of thea = 0 plane
VII0	0	1	1	0	A	has Euclidean space as a special case	first quadrant of the horizontal plane
VIIa	${\displaystyle a}$	0	1	1	B	has a hyper-pseudosphere as a special case	projects to first quadrant of thea = 0 plane
VIII	0	1	1	-1	A		sixth octant
IX	0	1	1	1	A	has ahypersphere as a special case	second octant

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 matrixC^ab:
   ${\displaystyle C_{ab}^c={\epsilon }_{abd}{C}^{dc},}$
   where ε_abd is the totally antisymmetric three-dimensional Levi-Civita symbol (ε₁₂₃ = 1). Substitution of this expression for${\displaystyle C_{ab}^c}$ into theJacobi identity, results in
   ${\displaystyle {\epsilon }_{abd}{C}^{bd}{C}^{ac}=0.}$
2. The structure constants can be transformed as:
   ${\displaystyle C^{ab}={\left(detA\right)}^{-1}{A}_m^a{A}_n^b{\acute{C}}^{mn}.}$
   Appearance of detA in this formula is due to the fact that the symbol ε_abd transforms astensor density:${\displaystyle {\epsilon }_{abc}=\left(detA\right)D_a^m{D}_b^n{D}_c^d{\acute{\epsilon }}_{mnd}}$, where έ_mnd ≡ ε_mnd. By this transformation it is always possible to reduce the matrixC^ab to the form:
   
   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. Ifn₁ ≠ 0 thenC²³ –C³² = 0 and by the remaining transformations with${\displaystyle A_{\overline{b}}^{\overline{a}}\ne 0,\overline{a},\overline{b}=\overline{2},\overline{3}}$, the 2 × 2 matrix${\displaystyle C^{\overline{a}\overline{b}}}$ inC^ab can be made diagonal. Then
   
   The diagonality condition forC^ab is preserved under the transformations with diagonal${\displaystyle A_b^a}$. Under these transformations, the three parametersn₁,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 anyn_a (if it is not zero) can be made equal to unity. Taking into account that the simultaneous change of sign of alln_a produce nothing new, one arrives to the following invariantly different sets for the numbersn₁,n₂,n₃ (invariantly different in the sense that there is no way to pass from one to another by some transformation of the triad${\displaystyle e_a^{\overline{a}}=A_{\overline{b}}^{\overline{a}}{e}_a^{\overline{b}}}$), that is to the following different types of homogeneous spaces with diagonal matrixC^ab:
   
5. Consider now the casen₁ = 0. It can also happen in that case thatC²³ –C³² = 0. This returns to the situation already analyzed in the previous step but with the additional conditionn₁ = 0. Now, all essentially different types for the setsn₁,n₂,n₃ are (0, 1, 1), (0, 1, −1), (0, 0, 1) and (0, 0, 0). The first three repeat the typesVII₀,VI₀,II. Consequently, only one new type arises:
   ${\displaystyle BianchiI:(n_1,n_2,n_3)=(0,0,0).}$
6. The only case left isn₁ = 0 andC²³ –C³² ≠ 0. Now the 2 × 2 matrix${\displaystyle C^{\overline{a}\overline{b}}(\overline{a},\overline{b}=2,3)}$ is non-symmetric and it cannot be made diagonal by transformations using${\displaystyle A_{\overline{b}}^{\overline{a}}\ne 0}$. However, its symmetric part can be diagonalized, that is the 3 × 3 matrixC^ab can be reduced to the form:
   
   wherea is an arbitrary number. After this is done, there still remains the possibility to perform transformations with diagonal${\displaystyle A_{\overline{b}}^{\overline{a}}}$, under which the quantitiesn₂,n₃ anda change as follows:
   ${\displaystyle n_2={\left(A_1^1{A}_2^2{A}_3^3\right)}^{-1}{\left(A_2^2\right)}^2{\acute{n}}_2,n_3={\left(A_1^1{A}_2^2{A}_3^3\right)}^{-1}{\left(A_3^3\right)}^2{\acute{n}}_3,a={\left(A_1^1\right)}^{-1}\acute{a}.}$
   These formulas show that for nonzeron₂,n₃,a, the combinationa²(n₂n₃)⁻¹ is an invariant quantity. By a choice of${\displaystyle A_1^1}$, one can impose the conditiona > 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 ofn₂ andn₃ 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 numbera 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 anda remains an invariant and arbitrary positive number. Historically these four types of homogeneous spaces have been classified as:
   
   TypeIII is just a particular case of typeVI corresponding toa = 1. TypesVII andVI contain an infinity of invariantly different types of algebras corresponding to the arbitrariness of the continuous parametera. TypeVII₀ is a particular case ofVII corresponding toa = 0 while typeVI₀ is a particular case ofVI corresponding also toa = 0.

The Bianchi spaces have the property that theirRicci tensors can beseparated into a product of thebasis vectors associated with the space and a coordinate-independent tensor.

For a givenmetric:

${\displaystyle d{s}^2={\gamma }_{ab}{\xi }_i^{(a)}{\xi }_k^{(b)}d{x}^{i}d{x}^k}$

(where${\displaystyle {\xi }_i^{(a)}d{x}^i}$　are1-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${\displaystyle x^i}$.

Incosmology, this classification is used for ahomogeneousspacetime 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. TheFriedmann–Lemaître–Robertson–Walker metrics are isotropic, which are particular cases of types I, V,${\displaystyle {\text{VII}}_h}$ and IX. The Bianchi type I models include theKasner metric as a special case.The Bianchi IX cosmologies include theTaub metric.^[2] However, the dynamics near the singularity is approximately governed by a series of successive Kasner (Bianchi I) periods. The complicated dynamics,which essentially amounts to billiard motion in a portion ofhyperbolic space, exhibits chaotic behaviour, and is namedMixmaster; its analysis is referred to as theBKL analysis after Belinskii, Khalatnikov and Lifshitz.^[3][4]More recent work has established a relation of (super-)gravity theories near a spacelike singularity (BKL-limit) with LorentzianKac–Moody algebras,Weyl groups and hyperbolicCoxeter groups.^[5][6][7]Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.^[8][9][10] In a space that is both homogeneous and isotropic the metric is determined completely, leaving free only the sign of the curvature. Assuming only space homogeneity with no additional symmetry such as isotropy leaves considerably more freedom in choosing the metric. The following pertains to the space part of the metric at a given instant of timet assuming a synchronous frame so thatt is the same synchronised time for the whole space.

Homogeneity implies identical metric properties at all points of the space. An exact definition of this concept involves considering sets of coordinate transformations that transform the space into itself, i.e. leave its metric unchanged: if theline element before transformation is

${\displaystyle d{l}^2={\gamma }_{\alpha \beta }\left(x^1,x^2,x^3\right)d{x}^{\alpha }d{x}^{\beta },}$

then after transformation the same line element is

${\displaystyle d{l}^2={\gamma }_{\alpha \beta }\left(x^{\mathrm{\prime }1},x^{\mathrm{\prime }2},x^{\mathrm{\prime }3}\right)d{x}^{\mathrm{\prime }\alpha }d{x}^{\mathrm{\prime }\beta },}$

with the same functional dependence of γ_αβ on the new coordinates. (For a more theoretical and coordinate-independent definition of homogeneous space seehomogeneous space). A space is homogeneous if it admits a set of transformations (a group of motions) that brings any given point to the position of any other point. Since space is three-dimensional the different transformations of the group are labelled by three independent parameters.

InEuclidean space the homogeneity of space is expressed by theinvariance of the metric under parallel displacements (translations) of theCartesian coordinate system. Each translation is determined by three parameters — the components of the displacement vector of the coordinate origin. All these transformations leave invariant the three independent differentials (dx,dy,dz) from which the line element is constructed. In the general case of a non-Euclidean homogeneous space, the transformations of its group of motions again leave invariant three independent lineardifferential forms, which do not, however, reduce tototal differentials of any coordinate functions. These forms are written as${\displaystyle e_{\alpha }^{(a)}d{x}^{\alpha }}$ where the Latin index (a) labels three independent vectors (coordinate functions); these vectors are called aframe field or triad. The Greek letters label the three space-likecurvilinear coordinates. A spatial metric invariant is constructed under the given group of motions with the use of the above forms:

${\displaystyle d{l}^2={\eta }_{ab}\left(e_{\alpha }^{(a)}d{x}^{\alpha }\right)\left(e_{\beta }^{(b)}d{x}^{\beta }\right)}$

eq. 6a

i.e. the metric tensor is

${\displaystyle {\gamma }_{\alpha \beta }={\eta }_{ab}{e}_{\alpha }^{(a)}{e}_{\beta }^{(b)}}$

eq. 6b

where the coefficients η_ab, which are symmetric in the indicesa andb, are functions of time. The choice of basis vectors is dictated by the symmetry properties of the space and, in general, these basis vectors are not orthogonal (so that the matrix η_ab is not diagonal).

The reciprocal triple of vectors${\displaystyle e_{(a)}^{\alpha }}$ is introduced with the help ofKronecker delta

${\displaystyle e_{(a)}^{\alpha }{e}_{\alpha }^{(b)}={\delta }_a^b;e_{(a)}^{\alpha }{e}_{\beta }^{(a)}={\delta }_{\alpha }^{\beta }}$

eq. 6c

In the three-dimensional case, the relation between the two vector triples can be written explicitly

${\displaystyle {\mathbf{e}}_{(1)}=\frac{1}{v}{\mathbf{e}}^{(2)}\times {\mathbf{e}}^{(3)},{\mathbf{e}}_{(2)}=\frac{1}{v}{\mathbf{e}}^{(3)}\times {\mathbf{e}}^{(1)},{\mathbf{e}}_{(3)}=\frac{1}{v}{\mathbf{e}}^{(1)}\times {\mathbf{e}}^{(2)},}$

eq. 6d

where the volumev is

${\displaystyle v=\left|e_{\alpha }^{(a)}\right|={\mathbf{e}}^{(1)}\cdot {\mathbf{e}}^{(2)}\times {\mathbf{e}}^{(3)},}$

withe_(a) ande^(a) regarded as Cartesian vectors with components${\displaystyle e_{(a)}^{\alpha }}$ and${\displaystyle e_{\alpha }^{(a)}}$, respectively. Thedeterminant of the metric tensoreq. 6b is γ = ηv² where η is the determinant of the matrix η_ab.

The required conditions for the homogeneity of the space are

${\displaystyle \left(\frac{\mathrm{\partial }{e}_{\alpha }^{(c)}}{\mathrm{\partial }{x}^{\beta }}-\frac{\mathrm{\partial }{e}_{\beta }^{(c)}}{\mathrm{\partial }{x}^{\alpha }}\right)e_{(a)}^{\alpha }{e}_{(b)}^{\beta }=C_{ab}^c.}$

eq. 6e

The constants${\displaystyle C_{ab}^c}$ are called thestructure constants of the group.

Proof ofeq. 6e

The invariance of the differential forms${\displaystyle e_{\alpha }^{(a)}d{x}^{\alpha }}$ means that ${\displaystyle e_{\alpha }^{(a)}(x)d{x}^{\alpha }=e_{\alpha }^{(a)}(x^{\mathrm{\prime }})d{x}^{\mathrm{\prime }\alpha }}$ where the${\displaystyle e_{\alpha }^{(a)}}$ on the two sides of the equation are the same functions of the old and new coordinates, respectively. Multiplying this equation by${\displaystyle e_{(a)}^{\beta }(x^{\mathrm{\prime }})}$, setting${\displaystyle d{x}^{\mathrm{\prime }\beta }=\frac{\mathrm{\partial }{x}^{\mathrm{\prime }\beta }}{\mathrm{\partial }{x}^{\alpha }}d{x}^{\alpha },}$ and comparing coefficients of the same differentialsdxα, one finds ${\displaystyle \frac{\mathrm{\partial }{x}^{\mathrm{\prime }\beta }}{\mathrm{\partial }{x}^{\alpha }}=e_{(a)}^{\beta }(x^{\mathrm{\prime }})e_{\alpha }^{(a)}(x).}$ These equations are a system of differential equations that determine the functions${\displaystyle x^{\mathrm{\prime }\beta }(x)}$ for a given frame. In order to be integrable, these equations must satisfy identically the conditions ${\displaystyle \frac{{\mathrm{\partial }}^2{x}^{\mathrm{\prime }\beta }}{\mathrm{\partial }{x}^{\alpha }\mathrm{\partial }{x}^{\gamma }}=\frac{{\mathrm{\partial }}^2{x}^{\mathrm{\prime }\beta }}{\mathrm{\partial }{x}^{\gamma }\mathrm{\partial }{x}^{\alpha }}.}$ Calculating the derivatives, one finds ${\displaystyle \left[\frac{\mathrm{\partial }{e}_{(a)}^{\beta }(x^{\mathrm{\prime }})}{\mathrm{\partial }{x}^{\mathrm{\prime }\delta }}{e}_{(b)}^{\delta }(x^{\mathrm{\prime }})-\frac{\mathrm{\partial }{e}_{(b)}^{\beta }(x^{\mathrm{\prime }})}{\mathrm{\partial }{x}^{\mathrm{\prime }\delta }}{e}_{(a)}^{\delta }(x^{\mathrm{\prime }})\right]e_{\gamma }^{(b)}(x)e_{\alpha }^{(a)}(x)=e_{(a)}^{\beta }(x^{\mathrm{\prime }})\left[\frac{\mathrm{\partial }{e}_{\gamma }^{(a)}(x)}{\mathrm{\partial }{x}^{\alpha }}-\frac{\mathrm{\partial }{e}_{\alpha }^{(a)}(x)}{\mathrm{\partial }{x}^{\gamma }}\right].}$ Multiplying both sides of the equations by${\displaystyle e_{(d)}^{\alpha }(x)e_{(c)}^{\gamma }(x)e_{\beta }^{(f)}(x^{\mathrm{\prime }})}$ and shifting the differentiation from one factor to the other by usingeq. 6c, one gets for the left side: ${\displaystyle e_{\beta }^{(f)}(x^{\mathrm{\prime }})\left[\frac{\mathrm{\partial }{e}_{(d)}^{\beta }(x^{\mathrm{\prime }})}{\mathrm{\partial }{x}^{\mathrm{\prime }\delta }}{e}_{(c)}^{\delta }(x^{\mathrm{\prime }})-\frac{\mathrm{\partial }{e}_{(c)}^{\beta }(x^{\mathrm{\prime }})}{\mathrm{\partial }{x}^{\mathrm{\prime }\delta }}{e}_{(d)}^{\delta }(x^{\mathrm{\prime }})\right]=e_{(c)}^{\beta }(x^{\mathrm{\prime }})e_{(d)}^{\delta }(x^{\mathrm{\prime }})\left[\frac{\mathrm{\partial }{e}_{\beta }^{(f)}(x^{\mathrm{\prime }})}{\mathrm{\partial }{x}^{\mathrm{\prime }\delta }}-\frac{\mathrm{\partial }{e}_{\delta }^{(f)}(x^{\mathrm{\prime }})}{\mathrm{\partial }{x}^{\mathrm{\prime }\beta }}\right].}$ and for the right, the same expression in the variablex. Sincex andx' are arbitrary, these expression must reduce to constants to obtaineq. 6e.

Multiplying by${\displaystyle e_{(c)}^{\gamma }}$,eq. 6e can be rewritten in the form

${\displaystyle e_{(a)}^{\alpha }\frac{\mathrm{\partial }{e}_{(b)}^{\gamma }}{\mathrm{\partial }{x}^{\alpha }}-e_{(b)}^{\beta }\frac{\mathrm{\partial }{e}_{(a)}^{\gamma }}{\mathrm{\partial }{x}^{\beta }}=C_{ab}^c{e}_{(c)}^{\gamma }.}$

eq. 6f

Equation 6e can be written in a vector form as

${\displaystyle {\mathbf{e}}_{(a)}\times {\mathbf{e}}_{(b)}\text{curl }{\mathbf{e}}^{(c)}=-C_{ab}^c,}$

where again the vector operations are done as if the coordinatesx^α were Cartesian. Usingeq. 6d, one obtains

${\displaystyle \frac{1}{v}\left({\mathbf{e}}^{(1)}\text{curl }{\mathbf{e}}^{(1)}\right)=C_{32}^1,\frac{1}{v}\left({\mathbf{e}}^{(2)}\text{curl }{\mathbf{e}}^{(1)}\right)=C_{13}^1,\frac{1}{v}\left({\mathbf{e}}^{(3)}\text{curl }{\mathbf{e}}^{(1)}\right)=C_{21}^1}$

eq. 6g

and six more equations obtained by a cyclic permutation of indices 1, 2, 3.

The structure constants are antisymmetric in their lower indices as seen from their definitioneq. 6e:${\displaystyle C_{ab}^c=-C_{ba}^c}$. Another condition on the structure constants can be obtained by noting thateq. 6f can be written in the form ofcommutation relations

${\displaystyle \left[X_a,X_b\right]\equiv X_a{X}_b-X_b{X}_a=C_{ab}^c{X}_c}$

eq. 6h

for the lineardifferential operators

${\displaystyle X_a=e_{(a)}^{\alpha }\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\alpha }}}$

eq. 6i

In the mathematical theory of continuous groups (Lie groups) the operatorsX_a satisfying conditionseq. 6h are called thegenerators of the group. The theory of Lie groups uses operators defined using theKilling vectors${\displaystyle {\xi }_{(a)}^{\alpha }}$ instead of triads${\displaystyle e_{(a)}^{\alpha }}$. Since in the synchronous metric none of the γ_αβ components depends on time, the Killing vectors (triads) are time-like.

The conditionseq. 6h follow from theJacobi identity

${\displaystyle [[X_a,X_b],X_c]+[[X_b,X_c],X_a]+[[X_c,X_a],X_b]=0}$

and have the form

${\displaystyle C_{ab}^e{C}_{ec}^d+C_{bc}^e{C}_{ea}^d+C_{ca}^e{C}_{eb}^d=0}$

eq. 6j

It is a definite advantage to use, in place of the three-index constants${\displaystyle C_{ab}^c}$, a set of two-index quantities, obtained by the dual transformation

${\displaystyle C_{ab}^c=e_{abd}{C}^{dc}}$

eq. 6k

wheree_abc =e^abc is theunit antisymmetric symbol (withe₁₂₃ = +1). With these constants the commutation relationseq. 6h are written as

${\displaystyle e^{abc}{X}_b{X}_c=C^{ad}{X}_d}$

eq. 6l

The antisymmetry property is already taken into account in the definitioneq. 6k, while propertyeq. 6j takes the form

${\displaystyle e_{bcd}{C}^{cd}{C}^{ba}=0}$

eq. 6m

The choice of the three frame vectors in the differential forms${\displaystyle e_{\alpha }^{(a)}d{x}^{\alpha }}$ (and with them the operatorsX_a) is not unique. They can be subjected to any linear transformation with constant coefficients:

${\displaystyle {\mathbf{e}}_{(a)}=A_a^b{\mathbf{e}}_{(b)}.}$

eq. 6n

The quantities η_ab andC^ab behave like tensors (are invariant) with respect to such transformations.

The conditionseq. 6m are the only ones that the structure constants must satisfy. But among the constants admissible by these conditions, there are equivalent sets, in the sense that their difference is related to a transformation of the typeeq. 6n. The question of the classification of homogeneous spaces reduces to determining all nonequivalent sets of structure constants. This can be done, using the "tensor" properties of the quantitiesC^ab, by the following simple method (C. G. Behr, 1962).

The asymmetric tensorC^ab can be resolved into a symmetric and an antisymmetric part. The first is denoted byn^ab, and the second is expressed in terms of itsdual vectora_c:

${\displaystyle C^{ab}=n^{ab}+e^{abc}{a}_c.}$

eq. 6o

Substitution of this expression ineq. 6m leads to the condition

${\displaystyle n^{ab}{a}_b=0.}$

eq. 6p

By means of the transformationseq. 6n the symmetric tensorn^ab can be brought to diagonal form witheigenvaluesn₁,n₂,n₃. Equation6p shows that the vectora_b (if it exists) lies along one of the principal directions of the tensorn^ab, the one corresponding to the eigenvalue zero. Without loss of generality one can therefore seta_b = (a, 0, 0). Theneq. 6p reduces toan₁ = 0, i.e. one of the quantitiesa orn₁ must be zero. The Jacobi identities take the form:

${\displaystyle [X_1,X_2]=-a{X}_2+n_3{X}_3,[X_2,X_3]=n_1{X}_1,[X_3,X_1]=n_2{X}_2+a{X}_3.}$

eq. 6q

The only remaining freedoms are sign changes of the operatorsX_a and their multiplication by arbitrary constants. This permits to simultaneously change the sign of all then_a and also to make the quantitya positive (if it is different from zero). Also all structure constants can be made equal to ±1, if at least one of the quantitiesa,n₂,n₃ vanishes. But if all three of these quantities differ from zero, the scale transformations leave invariant the ratioh =a²(n₂n₃)⁻¹.

Thus one arrives at the Bianchi classification listing the possible types of homogeneous spaces classified by the values ofa,n₁,n₂,n₃ which is graphically presented in Fig. 3. In the class A case (a = 0),type IX (n⁽¹⁾=1,n⁽²⁾=1,n⁽³⁾=1) is represented by octant 2,type VIII (n⁽¹⁾=1,n⁽²⁾=1,n⁽³⁾=–1) is represented by octant 6, whiletype VII₀ (n⁽¹⁾=1,n⁽²⁾=1,n⁽³⁾=0) is represented by the first quadrant of the horizontal plane andtype VI₀ (n⁽¹⁾=1,n⁽²⁾=–1,n⁽³⁾=0) is represented by the fourth quadrant of this plane;type II ((n⁽¹⁾=1,n⁽²⁾=0,n⁽³⁾=0) is represented by the interval [0,1] alongn⁽¹⁾ andtype I (n⁽¹⁾=0,n⁽²⁾=0,n⁽³⁾=0) is at the origin. Similarly in the class B case (withn⁽³⁾ = 0), Bianchitype VI_h (a=h,n⁽¹⁾=1,n⁽²⁾=–1) projects to the fourth quadrant of the horizontal plane andtype VII_h (a=h,n⁽¹⁾=1,n⁽²⁾=1) projects to the first quadrant of the horizontal plane; these last two types are a single isomorphism class corresponding to a constant value surface of the functionh =a²(n⁽¹⁾n⁽²⁾)⁻¹. A typical such surface is illustrated in one octant, the angleθ given by tan θ = |h/2|^1/2; those in the remaining octants are obtained by rotation through multiples ofπ/2,h alternating in sign for a given magnitude |h|.Type III is a subtype of VI_h witha=1.Type V (a=1,n⁽¹⁾=0,n⁽²⁾=0) is the interval (0,1] along the axisa andtype IV (a=1,n⁽¹⁾=1,n⁽²⁾=0) is the vertical open face between the first and fourth quadrants of thea = 0 plane with the latter giving the class A limit of each type.

The Einstein equations for a universe with a homogeneous space can reduce to a system of ordinary differential equations containing only functions of time with the help of a frame field. To do this one must resolve the spatial components of four-vectors and four-tensors along the triad of basis vectors of the space:

${\displaystyle R_{(a)(b)}=R_{\alpha \beta }{e}_{(a)}^{\alpha }{e}_{(b)}^{\beta },R_{0(a)}=R_{0\alpha }{e}_{(a)}^{\alpha },u^{(a)}=u^{\alpha }{e}_{\alpha }^{(a)},}$

where all these quantities are now functions oft alone; the scalar quantities, the energy density ε and the pressure of the matterp, are also functions of the time.

The Einstein equations in vacuum in synchronous reference frame are^{[11][12][note 1]}

${\displaystyle R_0^0=-\frac{1}{2}\frac{\mathrm{\partial }{\varkappa }_{\alpha }^{\alpha }}{\mathrm{\partial }t}-\frac{1}{4}{\varkappa }_{\alpha }^{\beta }{\varkappa }_{\beta }^{\alpha }=0,}$

eq. 11

${\displaystyle R_{\alpha }^{\beta }=-\frac{1}{2\sqrt{-g}}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\sqrt{-g}{\varkappa }_{\alpha }^{\beta }\right)-P_{\alpha }^{\beta }=0,}$

eq. 12

${\displaystyle R_{\alpha }^0=\frac{1}{2}\left({\varkappa }_{\alpha ;\beta }^{\beta }-{\varkappa }_{\beta ;\alpha }^{\beta }\right)=0,}$

eq. 13

where${\displaystyle {\varkappa }_{\alpha }^{\beta }}$ is the 3-dimensional tensor${\displaystyle {\varkappa }_{\alpha }^{\beta }=\frac{\mathrm{\partial }{\gamma }_{\alpha }^{\beta }}{\mathrm{\partial }t}}$, andP_αβ is the 3-dimensionalRicci tensor, which is expressed by the 3-dimensionalmetric tensor γ_αβ in the same way asR_ik is expressed byg_ik;P_αβ contains only the space (but not the time) derivatives of γ_αβ. Using triads, foreq. 11 one has simply

${\displaystyle {\varkappa }_{(a)(b)}=\frac{\mathrm{\partial }{\eta }_{ab}}{\mathrm{\partial }t},{\varkappa }_{(a)}^{(b)}=\frac{\mathrm{\partial }{\eta }_{ac}}{\mathrm{\partial }t}{\eta }^{cb}.}$

The components ofP_(a)(b) can be expressed in terms of the quantities η_ab and the structure constants of the group by using the tetrad representation of the Ricci tensor in terms of quantities${\displaystyle {\lambda }_{abc}=\left(e_{(a)i,k}-e_{(a)k,i}\right)e_{(b)}^i{e}_{(c)}^k}$^[13]

${\displaystyle R_{(a)(b)}=-\frac{1}{2}\left({\lambda }_{ab,c}^c+{\lambda }_{ba,c}^c+{\lambda }_{ca,b}^c+{\lambda }_{cb,a}^c+{\lambda }_b^{cd}{\lambda }_{cda}+{\lambda }_b^{cd}{\lambda }_{dca}-\frac{1}{2}{\lambda }_b^{cd}{\lambda }_{acd}+{\lambda }_{cd}^c{\lambda }_{ab}^d+{\lambda }_{cd}^c{\lambda }_{ba}^d\right).}$

After replacing the three-index symbols${\displaystyle {\lambda }_{bc}^a=C_{bc}^a}$ by two-index symbolsC^ab and the transformations:

${\displaystyle {\eta }_{ad}{\eta }_{bc}{\eta }_{cf}{e}^{def}=\eta e_{abc},e_{abf}{e}^{cdf}={\delta }_a^c{\delta }_b^d-{\delta }_a^d{\delta }_b^c}$

one gets the "homogeneous" Ricci tensor expressed in structure constants:

${\displaystyle P_{(a)}^{(b)}=\frac{1}{2\eta }\left\{2C^{bd}{C}_{ad}+C^{db}{C}_{ad}+C^{bd}{C}_{da}-C_d^d\left(C_a^b+C_a^b\right)+{\delta }_a^b\left[{\left(C_d^d\right)}^2-2C^{df}{C}_{df}\right]\right\}.}$

Here, all indices are raised and lowered with the local metric tensor η_ab

${\displaystyle C_a^b={\eta }_{ac}{C}^{cb},C_{ab}={\eta }_{ac}{\eta }_{bd}{C}^{cd}.}$

TheBianchi identities for the three-dimensional tensorP_αβ in the homogeneous space take the form

${\displaystyle P_b^c{C}_{ca}^b+P_a^c{C}_{cb}^b=0.}$

Taking into account the transformations of covariant derivatives for arbitrary four-vectorsA_i and four-tensorsA_ik

${\displaystyle A_{i;k}{e}_{(a)}^i{e}_{(b)}^k=A_{(a)(b)}-A^{(d)}{\gamma }_{dab},}$

${\displaystyle A_{ik;l}{e}_{(a)}^i{e}_{(b)}^k{e}_{(c)}^l=A_{(a)(b)(c)}-A_{(b)}^{(d)}{\gamma }_{dac}+A_{(a)}^{(d)}{\gamma }_{dbc},}$

the final expressions for the triad components of the Ricci four-tensor are:

${\displaystyle R_0^0=-\frac{1}{2}\frac{\mathrm{\partial }{\varkappa }_{(a)}^{(a)}}{\mathrm{\partial }t}-\frac{1}{4}{\varkappa }_{(a)}^{(b)}{\varkappa }_{(b)}^{(a)},}$

eq. 11a

${\displaystyle R_{(a)}^{(b)}=-\frac{1}{2\sqrt{\eta }}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\sqrt{\eta }{\varkappa }_{(a)}^{(b)}\right)-P_{(a)}^{(b)},}$

eq. 12a

${\displaystyle R_{(a)}^0=-\frac{1}{2}{\varkappa }_{(b)}^{(c)}\left(C_{ca}^b-{\delta }_a^b{C}_{dc}^d\right).}$

eq. 13a

In setting up the Einstein equations there is thus no need to use explicit expressions for the basis vectors as functions of the coordinates.

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