Bianchi classification

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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification_in_dimension_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Classification in dimension 3</span> </div> </a> <button aria-controls="toc-Classification_in_dimension_3-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Classification in dimension 3 subsection</span> </button> <ul id="toc-Classification_in_dimension_3-sublist" class="vector-toc-list"> <li id="toc-Structure_constants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structure_constants"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Structure constants</span> </div> </a> <ul id="toc-Structure_constants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curvature_of_Bianchi_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curvature_of_Bianchi_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Curvature of Bianchi spaces</span> </div> </a> <ul id="toc-Curvature_of_Bianchi_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cosmological_application" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cosmological_application"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Cosmological application</span> </div> </a> <ul id="toc-Cosmological_application-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Lie algebra classification</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Bianchi classification</b> provides a list of all real 3-dimensional <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> (<a href="/wiki/Up_to" title="Up to">up to</a> <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>). 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 associated <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> serve as symmetry groups of 3-dimensional <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifolds</a>. It is named for <a href="/wiki/Luigi_Bianchi" title="Luigi Bianchi">Luigi Bianchi</a>, who worked it out in 1898. </p><p>The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of <a href="/wiki/Complex_Lie_algebra" title="Complex Lie algebra">complex Lie algebras</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Classification_in_dimension_less_than_3">Classification in dimension less than 3</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=1" title="Edit section: Classification in dimension less than 3"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Dimension 0: The only Lie algebra is the <a href="/wiki/Abelian_Lie_algebra" class="mw-redirect" title="Abelian Lie algebra">abelian Lie algebra</a> <b>R</b><sup>0</sup>.</li> <li>Dimension 1: The only Lie algebra is the abelian Lie algebra <b>R</b><sup>1</sup>, with <a href="/wiki/Outer_automorphism_group" title="Outer automorphism group">outer automorphism group</a> the multiplicative group of non-zero real numbers.</li> <li>Dimension 2: There are two Lie algebras: <ul><li>(1) The abelian Lie algebra <b>R</b><sup>2</sup>, with outer automorphism group <a href="/wiki/General_linear_group" title="General linear group">GL<sub>2</sub>(<b>R</b>)</a>.</li> <li>(2) The <a href="/wiki/Solvable_Lie_algebra" title="Solvable Lie algebra">solvable Lie algebra</a> of 2×2 upper triangular matrices of trace 0. It has trivial center and trivial outer automorphism group. The <a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">associated</a> <a href="/wiki/Simply_connected_space" title="Simply connected space">simply connected</a> <a href="/wiki/Lie_group" title="Lie group">Lie group</a> is the <a href="/wiki/Affine_group" title="Affine group">affine group</a> of the line.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Classification_in_dimension_3">Classification in dimension 3</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=2" title="Edit section: Classification in dimension 3"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as a <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of <b>R</b><sup>2</sup> and <b>R</b>, with <b>R</b> acting on <b>R</b><sup>2</sup> by some 2 by 2 matrix <i>M</i>. The different types correspond to different types of matrices <i>M</i>, as described below. </p> <ul><li><b>Type I</b>: This is the abelian and unimodular Lie algebra <b>R</b><sup>3</sup>. The simply connected group has center <b>R</b><sup>3</sup> and outer automorphism group GL<sub>3</sub>(<b>R</b>). This is the case when <i>M</i> is 0.</li> <li><b>Type II</b>: The <a href="/wiki/Heisenberg_algebra" class="mw-redirect" title="Heisenberg algebra">Heisenberg algebra</a>, which is <a href="/wiki/Nilpotent_Lie_algebra" title="Nilpotent Lie algebra">nilpotent</a> and unimodular. The simply connected group has center <b>R</b> and outer automorphism group GL<sub>2</sub>(<b>R</b>). This is the case when <i>M</i> is nilpotent but not 0 (eigenvalues all 0).</li> <li><b>Type III</b>: This algebra is a product of <b>R</b> and the 2-dimensional non-abelian Lie algebra. (It is a limiting case of type VI, where one eigenvalue becomes zero.) It is <a href="/wiki/Solvable_Lie_algebra" title="Solvable Lie algebra">solvable</a> and not unimodular. The simply connected group has center <b>R</b> and outer automorphism group the group of non-zero real numbers. The matrix <i>M</i> has one zero and one non-zero eigenvalue.</li> <li><b>Type IV</b>: The algebra generated by [<i>y</i>,<i>z</i>] = 0, [<i>x</i>,<i>y</i>] = <i>y</i>, [<i>x</i>, <i>z</i>] = <i>y</i> + <i>z</i>. 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 matrix <i>M</i> has two equal non-zero eigenvalues, but is not <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a>.</li> <li><b>Type V</b>: [<i>y</i>,<i>z</i>] = 0, [<i>x</i>,<i>y</i>] = <i>y</i>, [<i>x</i>, <i>z</i>] = <i>z</i>. 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 GL<sub>2</sub>(<b>R</b>) of determinant +1 or −1. The matrix <i>M</i> has two equal eigenvalues, and is diagonalizable.</li> <li><b>Type VI</b>: An infinite family: semidirect products of <b>R</b><sup>2</sup> by <b>R</b>, where the matrix <i>M</i> 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.</li> <li><b>Type VI<sub>0</sub></b>: This Lie algebra is the semidirect product of <b>R</b><sup>2</sup> by <b>R</b>, with <b>R</b> where the matrix <i>M</i> has non-zero distinct real eigenvalues with zero sum. It is solvable and unimodular. It is the Lie algebra of the 2-dimensional <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>, the group of isometries of 2-dimensional <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>. The simply connected group has trivial center and outer automorphism group the product of the <a href="/wiki/Positive_real_numbers" title="Positive real numbers">positive real numbers</a> with the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> of order 8.</li> <li><b>Type VII</b>: An infinite family: semidirect products of <b>R</b><sup>2</sup> by <b>R</b>, where the matrix <i>M</i> 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.</li> <li><b>Type VII<sub>0</sub></b>: Semidirect product of <b>R</b><sup>2</sup> by <b>R</b>, where the matrix <i>M</i> 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 center <b>Z</b> and outer automorphism group a product of the non-zero real numbers and a group of order 2.</li> <li><b>Type VIII</b>: The Lie algebra <i>sl</i><sub>2</sub>(<b>R</b>) of traceless 2 by 2 matrices, associated to the group <a href="/wiki/SL2(R)" title="SL2(R)">SL<sub>2</sub>(R)</a>. It is <a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">simple</a> and unimodular. The simply connected group is not a matrix group; it is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>SL</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ed9322e8f9616764e6630498e55386893cae2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.869ex; height:3.676ex;" alt="{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}"></span>, has center <b>Z</b> and its outer automorphism group has order 2.</li> <li><b>Type IX</b>: The Lie algebra of the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> <i>O</i><sub>3</sub>(<b>R</b>). It is denoted by <a href="/wiki/3D_rotation_group#Lie_algebra" title="3D rotation group">𝖘𝖔(3)</a> and is simple and unimodular. The corresponding simply connected group is <a href="/wiki/Special_unitary_group#The_group_SU(2)" title="Special unitary group">SU(2)</a>; it has center of order 2 and trivial outer automorphism group, and is a <a href="/wiki/Spin_group" title="Spin group">spin group</a>.</li></ul> <p>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. </p><p>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. </p><p>The groups are related to the 8 geometries of Thurston's <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a>. 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 type <i>S<sup>2</sup></i>×<b>R</b> cannot be realized in this way. </p> <div class="mw-heading mw-heading3"><h3 id="Structure_constants">Structure constants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=3" title="Edit section: Structure constants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The three-dimensional Bianchi spaces each admit a set of three <a href="/wiki/Killing_vector" class="mw-redirect" title="Killing vector">Killing vector fields</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{i}^{(a)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}^{(a)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdec9db16be3ccf403a04e04df6c44071a064749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.415ex; height:3.676ex;" alt="{\displaystyle \xi _{i}^{(a)}}"></span> which obey the following property: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8936c76622a9902f833a6c6ba80cbb2e0eb30e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.901ex; height:8.176ex;" alt="{\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}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\ ab}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\ ab}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94861bdd40305fabac1751e5e0e8f1aa182d873d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.05ex; height:2.843ex;" alt="{\displaystyle C_{\ ab}^{c}}"></span>, the "structure constants" of the group, form a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> <a href="/wiki/Tensor" title="Tensor">order-three tensor</a> <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric</a> in its lower two indices. For any three-dimensional Bianchi space, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\ ab}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\ ab}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94861bdd40305fabac1751e5e0e8f1aa182d873d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.05ex; height:2.843ex;" alt="{\displaystyle C_{\ ab}^{c}}"></span> is given by the relationship </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\ ab}^{c}=\varepsilon _{abd}n^{cd}-\delta _{a}^{c}a_{b}+\delta _{b}^{c}a_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>d</mi> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>−</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\ ab}^{c}=\varepsilon _{abd}n^{cd}-\delta _{a}^{c}a_{b}+\delta _{b}^{c}a_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8fe38c10bc9ded83ed99f0e619ad66d5875013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.409ex; height:3.343ex;" alt="{\displaystyle C_{\ ab}^{c}=\varepsilon _{abd}n^{cd}-\delta _{a}^{c}a_{b}+\delta _{b}^{c}a_{a}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{abd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{abd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dded4a6148abdacaf91cf5bdf4bbd653f051bb82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.75ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{abd}}"></span> is the <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{a}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f6d724ba4dbb27c4287cbe5674192c3005d2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.676ex;" alt="{\displaystyle \delta _{a}^{c}}"></span> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, and the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{a}=(a,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{a}=(a,0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed11b4908aa8067b5b426089923d05d534671aa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.862ex; height:2.843ex;" alt="{\displaystyle a_{a}=(a,0,0)}"></span> and <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal</a> tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{cd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{cd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b1adeb4d4e0f1c209beeeaec2db11ebf0a7122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.199ex; height:2.676ex;" alt="{\displaystyle n^{cd}}"></span> are described by the following table, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{(i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{(i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d04aeb31aa43b3f20d1a162952f14acf5cab59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.474ex; height:2.843ex;" alt="{\displaystyle n^{(i)}}"></span> gives the <i>i</i>th <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{cd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{cd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b1adeb4d4e0f1c209beeeaec2db11ebf0a7122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.199ex; height:2.676ex;" alt="{\displaystyle n^{cd}}"></span>;<sup id="cite_ref-FOOTNOTELandauLifshitz1988_1-0" class="reference"><a href="#cite_note-FOOTNOTELandauLifshitz1988-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the parameter <i>a</i> runs over all positive <a href="/wiki/Real_number" title="Real number">real numbers</a>: </p> <table class="wikitable" align="center"> <tbody><tr> <th>Bianchi type </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{(1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{(1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc2fec9eaa42aee56b769eb62902b332c4c6032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:2.843ex;" alt="{\displaystyle n^{(1)}}"></span> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13583c68e42591dbec65eb639f64b3a94b278ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:2.843ex;" alt="{\displaystyle n^{(2)}}"></span> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{(3)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{(3)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9899606793427224207dd0af00a0eec6b5647d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:2.843ex;" alt="{\displaystyle n^{(3)}}"></span> </th> <th>class </th> <th>notes </th> <th>graphical (Fig. 1) </th></tr> <tr> <td>I</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>A</td> <td>describes <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean space</a></td> <td>at the origin </td></tr> <tr> <td>II</td> <td>0</td> <td>1</td> <td>0</td> <td>0</td> <td>A</td> <td></td> <td>interval [0,1] along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{(1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{(1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc2fec9eaa42aee56b769eb62902b332c4c6032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:2.843ex;" alt="{\displaystyle n^{(1)}}"></span> </td></tr> <tr> <td>III</td> <td>1</td> <td>0</td> <td>1</td> <td>-1</td> <td>B</td> <td>the subcase of type VI<sub><i>a</i></sub> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6104442ed30596ef4d7795d3186273f68d796ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}"></span></td> <td>projects to fourth quadrant of the <i>a</i> = 0 plane </td></tr> <tr> <td>IV</td> <td>1</td> <td>0</td> <td>0</td> <td>1</td> <td>B</td> <td></td> <td>vertical open face between first and fourth quadrants of the <i>a</i> = 0 plane </td></tr> <tr> <td>V</td> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td>B</td> <td>has a hyper-<a href="/wiki/Pseudosphere" title="Pseudosphere">pseudosphere</a> as a special case</td> <td>the interval (0,1] along the axis <i>a</i> </td></tr> <tr> <td>VI<sub>0</sub></td> <td>0</td> <td>1</td> <td>-1</td> <td>0</td> <td>A</td> <td></td> <td>fourth quadrant of the horizontal plane </td></tr> <tr> <td>VI<sub><i>a</i></sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span></td> <td>0</td> <td>1</td> <td>-1</td> <td>B</td> <td>when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6104442ed30596ef4d7795d3186273f68d796ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}"></span>, equivalent to type III</td> <td>projects to fourth quadrant of the <i>a</i> = 0 plane </td></tr> <tr> <td>VII<sub>0</sub></td> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td>A</td> <td>has Euclidean space as a special case</td> <td>first quadrant of the horizontal plane </td></tr> <tr> <td>VII<sub><i>a</i></sub></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span></td> <td>0</td> <td>1</td> <td>1</td> <td>B</td> <td>has a hyper-pseudosphere as a special case</td> <td>projects to first quadrant of the <i>a</i> = 0 plane </td></tr> <tr> <td>VIII</td> <td>0</td> <td>1</td> <td>1</td> <td>-1</td> <td>A</td> <td></td> <td>sixth octant </td></tr> <tr> <td>IX</td> <td>0</td> <td>1</td> <td>1</td> <td>1</td> <td>A</td> <td>has a <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a> as a special case</td> <td>second octant </td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bianchi-classification.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Bianchi-classification.svg/250px-Bianchi-classification.svg.png" decoding="async" width="250" height="416" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Bianchi-classification.svg/375px-Bianchi-classification.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Bianchi-classification.svg/500px-Bianchi-classification.svg.png 2x" data-file-width="2076" data-file-height="3456" /></a><figcaption><b>Figure 1.</b> The parameter space as a 3-plane (class A) and an orthogonal half 3-plane (class B) in <i>R</i><sup>4</sup> with coordinates (<i>n</i><sup>(1)</sup>, <i>n</i><sup>(2)</sup>, <i>n</i><sup>(3)</sup>, <i>a</i>), showing the canonical representatives of each Bianchi type.</figcaption></figure> <p>The standard Bianchi classification can be derived from the structural constants in the following six steps: </p> <ol><li>Due to the antisymmetry <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}=-C_{ba}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}=-C_{ba}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a74adf5ba3375b3894ce6865a4347049bc28c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.845ex; height:2.843ex;" alt="{\displaystyle C_{ab}^{c}=-C_{ba}^{c}}"></span>, there are nine independent constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74defdb40853a227a7df13fe27c636dce3e2b189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.469ex; height:2.843ex;" alt="{\displaystyle C_{ab}^{c}}"></span>. These can be equivalently represented by the nine components of an arbitrary constant matrix <i>C</i><sup><i>ab</i></sup>: <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}=\varepsilon _{abd}C^{dc},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>d</mi> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}=\varepsilon _{abd}C^{dc},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad816a109d38c08d6473a3d5588123a44b97de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.567ex; height:3.343ex;" alt="{\displaystyle C_{ab}^{c}=\varepsilon _{abd}C^{dc},}"></span><br />where ε<sub><i>abd</i></sub> is the totally antisymmetric three-dimensional Levi-Civita symbol (ε<sub>123</sub> = 1). Substitution of this expression for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74defdb40853a227a7df13fe27c636dce3e2b189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.469ex; height:2.843ex;" alt="{\displaystyle C_{ab}^{c}}"></span> into the <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a>, results in <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{abd}C^{bd}C^{ac}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>d</mi> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>d</mi> </mrow> </msup> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{abd}C^{bd}C^{ac}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4a1522708bcb36f14f32e4e13124527302a6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.865ex; height:3.009ex;" alt="{\displaystyle \varepsilon _{abd}C^{bd}C^{ac}=0.}"></span></li> <li>The structure constants can be transformed as:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{ab}=\left(\det {A}\right)^{-1}A_{m}^{a}A_{n}^{b}{\acute {C}}^{mn}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{ab}=\left(\det {A}\right)^{-1}A_{m}^{a}A_{n}^{b}{\acute {C}}^{mn}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086a9aea7141caeec41ef824cdeec92a9d0fb905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.659ex; height:3.509ex;" alt="{\displaystyle C^{ab}=\left(\det {A}\right)^{-1}A_{m}^{a}A_{n}^{b}{\acute {C}}^{mn}.}"></span><br />Appearance of det <b>A</b> in this formula is due to the fact that the symbol ε<sub><i>abd</i></sub> transforms as tensor density: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{abc}=\left(\det {A}\right)D_{a}^{m}D_{b}^{n}D_{c}^{d}{\acute {\varepsilon }}_{mnd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{abc}=\left(\det {A}\right)D_{a}^{m}D_{b}^{n}D_{c}^{d}{\acute {\varepsilon }}_{mnd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060d08a8b7431dccc0a4131db08e45f9e7209685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.893ex; height:3.176ex;" alt="{\displaystyle \varepsilon _{abc}=\left(\det {A}\right)D_{a}^{m}D_{b}^{n}D_{c}^{d}{\acute {\varepsilon }}_{mnd}}"></span>, where έ<sub><i>mnd</i></sub> ≡ ε<sub><i>mnd</i></sub>. By this transformation it is always possible to reduce the matrix <i>C</i><sup><i>ab</i></sup> to the form:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&C^{22}&C^{23}\\0&C^{32}&C^{33}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&C^{22}&C^{23}\\0&C^{32}&C^{33}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6870700f5b2ffaa944eadc30d5ab876509939f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:25.645ex; height:9.509ex;" alt="{\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&C^{22}&C^{23}\\0&C^{32}&C^{33}\end{bmatrix}}.}"></span><br />After such a choice, one still have the freedom of making triad transformations but with the restrictions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}^{1}=A_{3}^{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}^{1}=A_{3}^{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570ac8c6552942460f496db6b6e7bcaee147e579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.954ex; height:3.176ex;" alt="{\displaystyle A_{2}^{1}=A_{3}^{1}=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}^{2}=A_{1}^{3}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}^{2}=A_{1}^{3}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/432f6c4e16f3ea6a60c8ed83bcc89430b10cc0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.601ex; height:3.176ex;" alt="{\displaystyle A_{1}^{2}=A_{1}^{3}=0.}"></span></li> <li>Now, the Jacobi identities give only one constraint:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(C^{23}-C^{32}\right)n_{1}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(C^{23}-C^{32}\right)n_{1}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717b19b0add0db4a2c72176e9e97b584e467d99c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.062ex; height:3.343ex;" alt="{\displaystyle \left(C^{23}-C^{32}\right)n_{1}=0.}"></span></li> <li>If <i>n</i><sub>1</sub> ≠ 0 then <i>C</i><sup>23</sup> – <i>C</i><sup>32</sup> = 0 and by the remaining transformations with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0,\quad {\bar {a}},{\bar {b}}={\bar {2}},{\bar {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msubsup> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0,\quad {\bar {a}},{\bar {b}}={\bar {2}},{\bar {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b7bcd24c35f07d2fe2012eab05d2079f92b5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.346ex; height:3.676ex;" alt="{\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0,\quad {\bar {a}},{\bar {b}}={\bar {2}},{\bar {3}}}"></span>, the 2 × 2 matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{{\bar {a}}{\bar {b}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{{\bar {a}}{\bar {b}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f671ebec46cb837ae425534c7a820cbc49e3492d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.722ex; height:3.009ex;" alt="{\displaystyle C^{{\bar {a}}{\bar {b}}}}"></span> in <i>C</i><sup><i>ab</i></sup> can be made diagonal. Then<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&n_{2}&0\\0&0&n_{3}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&n_{2}&0\\0&0&n_{3}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda3cf92373a30c7ebfec6023024494cff7e9d92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.195ex; height:9.176ex;" alt="{\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&n_{2}&0\\0&0&n_{3}\end{bmatrix}}.}"></span><br />The diagonality condition for <i>C</i><sup><i>ab</i></sup> is preserved under the transformations with diagonal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{b}^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{b}^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fad8d49764f986ba4e231a4a143412e6b88a373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.845ex; height:2.843ex;" alt="{\displaystyle A_{b}^{a}}"></span>. Under these transformations, the three parameters <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub> change in the following way:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>no summation over</mtext> </mrow> <mtext> </mtext> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a85ff48af7bd019f43f9b0ce3f78c63b6df1fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.4ex; height:3.509ex;" alt="{\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.}"></span><br />By these diagonal transformations, the modulus of any <i>n</i><sub><i>a</i></sub> (if it is not zero) can be made equal to unity. Taking into account that the simultaneous change of sign of all <i>n</i><sub><i>a</i></sub> produce nothing new, one arrives to the following invariantly different sets for the numbers <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub> (invariantly different in the sense that there is no way to pass from one to another by some transformation of the triad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{a}^{\bar {a}}=A_{\bar {b}}^{\bar {a}}e_{a}^{\bar {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{a}^{\bar {a}}=A_{\bar {b}}^{\bar {a}}e_{a}^{\bar {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b0a24e6ecfa99bc7fb4b8b9a9af1ae979a66d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.314ex; height:3.843ex;" alt="{\displaystyle e_{a}^{\bar {a}}=A_{\bar {b}}^{\bar {a}}e_{a}^{\bar {b}}}"></span>), that is to the following different types of homogeneous spaces with diagonal matrix <i>C</i><sup><i>ab</i></sup>:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>I</mi> <mi>X</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>V</mi> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>V</mi> <mi>I</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>V</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>I</mi> <mi>I</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e4431f7dd6f9fb74b6dcd9b71b077c0c4697f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:47.763ex; height:16.509ex;" alt="{\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}}}"></span></li> <li>Consider now the case <i>n</i><sub>1</sub> = 0. It can also happen in that case that <i>C</i><sup>23</sup> – <i>C</i><sup>32</sup> = 0. This returns to the situation already analyzed in the previous step but with the additional condition <i>n</i><sub>1</sub> = 0. Now, all essentially different types for the sets <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub> are (0, 1, 1), (0, 1, −1), (0, 0, 1) and (0, 0, 0). The first three repeat the types <i>VII</i><sub>0</sub>, <i>VI</i><sub>0</sub>, <i>II</i>. Consequently, only one new type arises:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Bianchi\ I\ :\ (n_{1},n_{2},n_{3})\ =\ (0,0,0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>I</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Bianchi\ I\ :\ (n_{1},n_{2},n_{3})\ =\ (0,0,0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304a22acad4ea15b64be86e3bc222f7972f22562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.685ex; height:2.843ex;" alt="{\displaystyle Bianchi\ I\ :\ (n_{1},n_{2},n_{3})\ =\ (0,0,0).}"></span></li> <li>The only case left is <i>n</i><sub>1</sub> = 0 and <i>C</i><sup>23</sup> – <i>C</i><sup>32</sup> ≠ 0. Now the 2 × 2 matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{{\bar {a}}{\bar {b}}}({\bar {a}},{\bar {b}}=2,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{{\bar {a}}{\bar {b}}}({\bar {a}},{\bar {b}}=2,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf790a0d93635d662b0ecd5b29a0e877dd743f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.414ex; height:3.509ex;" alt="{\displaystyle C^{{\bar {a}}{\bar {b}}}({\bar {a}},{\bar {b}}=2,3)}"></span> is non-symmetric and it cannot be made diagonal by transformations using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msubsup> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132f1b06cadc4447eb8c82c784e63bd3be613836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.106ex; height:3.676ex;" alt="{\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0}"></span>. However, its symmetric part can be diagonalized, that is the 3 × 3 matrix <i>C</i><sup><i>ab</i></sup> can be reduced to the form:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{ab}={\begin{bmatrix}0&0&0\\0&n_{2}&a\\0&-a&n_{3}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>a</mi> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{ab}={\begin{bmatrix}0&0&0\\0&n_{2}&a\\0&-a&n_{3}\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf98291f217aefa7857394f38e35227fa0d702b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.497ex; height:9.176ex;" alt="{\displaystyle C^{ab}={\begin{bmatrix}0&0&0\\0&n_{2}&a\\0&-a&n_{3}\end{bmatrix}},}"></span><br />where <i>a</i> is an arbitrary number. After this is done, there still remains the possibility to perform transformations with diagonal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\bar {b}}^{\bar {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\bar {b}}^{\bar {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80413c6272c779bee953f29933027352824951d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.845ex; height:3.676ex;" alt="{\displaystyle A_{\bar {b}}^{\bar {a}}}"></span>, under which the quantities <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub> and <i>a</i> change as follows:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fdc9039c78110d2b9b7fe7de77fa40d5ef0ddde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:73.843ex; height:3.676ex;" alt="{\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}}.}"></span><br />These formulas show that for nonzero <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub>, <i>a</i>, the combination <i>a</i><sup>2</sup>(<i>n</i><sub>2</sub><i>n</i><sub>3</sub>)<sup>−1</sup> is an invariant quantity. By a choice of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1b908aa854e53aaed0e344b19aedb5298d76a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle A_{1}^{1}}"></span>, one can impose the condition <i>a</i> > 0 and after this is done, the choice of the sign of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da241397c16f3618ac9a32e28d1a2ae9a4fea83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.057ex; height:3.676ex;" alt="{\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}"></span> permits one to change both signs of <i>n</i><sub>2</sub> and <i>n</i><sub>3</sub> simultaneously, that is the set (<i>n</i><sub>2</sub> , <i>n</i><sub>3</sub>) is equivalent to the set (−<i>n</i><sub>2</sub>,−<i>n</i><sub>3</sub>). It follows that there are the following four different possibilities:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,n_{2},n_{3})=(a,0,0),(a,0,1),(a,1,1),(a,1,-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,n_{2},n_{3})=(a,0,0),(a,0,1),(a,1,1),(a,1,-1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145b00c8e772cd8eef261d989493df5c2452ef79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.388ex; height:2.843ex;" alt="{\displaystyle (a,n_{2},n_{3})=(a,0,0),(a,0,1),(a,1,1),(a,1,-1).}"></span><br />For the first two, the number <i>a</i> can be transformed to unity by a choice of<br />the parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1b908aa854e53aaed0e344b19aedb5298d76a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle A_{1}^{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da241397c16f3618ac9a32e28d1a2ae9a4fea83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.057ex; height:3.676ex;" alt="{\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}"></span>. For the second two possibilities, both of these parameters are already fixed and <i>a</i> remains an invariant and arbitrary positive number. Historically these four types of homogeneous spaces have been classified as:<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>V</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>I</mi> <mi>V</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>V</mi> <mi>I</mi> <mi>I</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>i</mi> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mtext> </mtext> <mi>V</mi> <mi>I</mi> </mtd> <mtd> <mo>:</mo> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d057f79db1e2bb66fd723153b92c6874a73b849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:53.764ex; height:16.509ex;" alt="{\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}}}"></span><br />Type <i>III</i> is just a particular case of type <i>VI</i> corresponding to <i>a</i> = 1. Types <i>VII</i> and <i>VI</i> contain an infinity of invariantly different types of algebras corresponding to the arbitrariness of the continuous parameter <i>a</i>. Type <i>VII</i><sub>0</sub> is a particular case of <i>VII</i> corresponding to <i>a</i> = 0 while type <i>VI</i><sub>0</sub> is a particular case of <i>VI</i> corresponding also to <i>a</i> = 0.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Curvature_of_Bianchi_spaces">Curvature of Bianchi spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=4" title="Edit section: Curvature of Bianchi spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Bianchi spaces have the property that their <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensors</a> can be <a href="/wiki/Separation_of_variables" title="Separation of variables">separated</a> into a product of the <a href="/wiki/Basis_vector" class="mw-redirect" title="Basis vector">basis vectors</a> associated with the space and a coordinate-independent tensor. </p><p>For a given <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=\gamma _{ab}\xi _{i}^{(a)}\xi _{k}^{(b)}dx^{i}dx^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=\gamma _{ab}\xi _{i}^{(a)}\xi _{k}^{(b)}dx^{i}dx^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bee6df3b162c8aa7ab3756d561cc440b0403b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.116ex; height:3.676ex;" alt="{\displaystyle ds^{2}=\gamma _{ab}\xi _{i}^{(a)}\xi _{k}^{(b)}dx^{i}dx^{k}}"></span></dd></dl> <p>(where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{i}^{(a)}dx^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}^{(a)}dx^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20b36b393411201d366b26d5a6d6cd19fca85b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.76ex; height:3.676ex;" alt="{\displaystyle \xi _{i}^{(a)}dx^{i}}"></span>　are <a href="/wiki/Differential_form" title="Differential form">1-forms</a>), the Ricci curvature tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{ik}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{ik}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807622f848b7ff6c4975a72ca84d38bc0bcd0d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.42ex; height:2.509ex;" alt="{\displaystyle R_{ik}}"></span> is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{ik}=R_{(a)(b)}\xi _{i}^{(a)}\xi _{k}^{(b)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{ik}=R_{(a)(b)}\xi _{i}^{(a)}\xi _{k}^{(b)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5eeca454a5822a506272fead18e00a6f646f839" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.314ex; height:3.843ex;" alt="{\displaystyle R_{ik}=R_{(a)(b)}\xi _{i}^{(a)}\xi _{k}^{(b)}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mtext> </mtext> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>c</mi> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mtext> </mtext> <mi>d</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mtext> </mtext> <mi>d</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>c</mi> <mi>d</mi> </mrow> </msubsup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9e457a8e070f2e224534b4c9126b5383ca2ad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.241ex; height:6.176ex;" alt="{\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]}"></span></dd></dl> <p>where the indices on the structure constants are raised and lowered with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{ab}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{ab}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f59a39ad5d1f52f78ee0674592d72494596240fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.012ex; height:2.176ex;" alt="{\displaystyle \gamma _{ab}}"></span> which is not a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8cd88951c7c0e3b181f956531b0a878bbed203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle x^{i}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Cosmological_application">Cosmological application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=5" title="Edit section: Cosmological application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Cosmology" title="Cosmology">cosmology</a>, this classification is used for a <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous</a> <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> 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 <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker metrics</a> are isotropic, which are particular cases of types I, V, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{VII}}_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>VII</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{VII}}_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2eb19c47de8245dc8db360ce916d66d328808ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\text{VII}}_{h}}"></span> and IX. The Bianchi type I models include the <a href="/wiki/Kasner_metric" title="Kasner metric">Kasner metric</a> as a special case. The Bianchi IX cosmologies include the <a href="/wiki/Taub-NUT_vacuum" class="mw-redirect" title="Taub-NUT vacuum">Taub metric</a>.<sup id="cite_ref-FOOTNOTEWald1984_2-0" class="reference"><a href="#cite_note-FOOTNOTEWald1984-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 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 of hyperbolic space, exhibits chaotic behaviour, and is named <a href="/wiki/Mixmaster_universe" title="Mixmaster universe">Mixmaster</a>; its analysis is referred to as the <a href="/wiki/BKL_singularity" title="BKL singularity">BKL analysis</a> after Belinskii, Khalatnikov and Lifshitz.<sup id="cite_ref-FOOTNOTEBelinskyKhalatnikovLifshitz1971_3-0" class="reference"><a href="#cite_note-FOOTNOTEBelinskyKhalatnikovLifshitz1971-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBelinskyKhalatnikovLifshitz1972_4-0" class="reference"><a href="#cite_note-FOOTNOTEBelinskyKhalatnikovLifshitz1972-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> More recent work has established a relation of (super-)gravity theories near a spacelike singularity (BKL-limit) with Lorentzian <a href="/wiki/Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebras</a>, <a href="/wiki/Weyl_group" title="Weyl group">Weyl groups</a> and hyperbolic <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a>.<sup id="cite_ref-FOOTNOTEHenneauxPerssonSpindel2008_5-0" class="reference"><a href="#cite_note-FOOTNOTEHenneauxPerssonSpindel2008-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHenneauxPerssonWesley2008_6-0" class="reference"><a href="#cite_note-FOOTNOTEHenneauxPerssonWesley2008-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHenneaux2009_7-0" class="reference"><a href="#cite_note-FOOTNOTEHenneaux2009-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.<sup id="cite_ref-FOOTNOTECornishLevin1997a_8-0" class="reference"><a href="#cite_note-FOOTNOTECornishLevin1997a-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTECornishLevin1997b_9-0" class="reference"><a href="#cite_note-FOOTNOTECornishLevin1997b-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTECornishLevin1997c_10-0" class="reference"><a href="#cite_note-FOOTNOTECornishLevin1997c-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> 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 time <i>t</i> assuming a synchronous frame so that <i>t</i> is the same synchronised time for the whole space. </p><p>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 the line element before transformation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dl^{2}=\gamma _{\alpha \beta }\left(x^{1},x^{2},x^{3}\right)dx^{\alpha }dx^{\beta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dl^{2}=\gamma _{\alpha \beta }\left(x^{1},x^{2},x^{3}\right)dx^{\alpha }dx^{\beta },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b01b00ae2ef3d68ae0aaef266c18f311395954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.812ex; height:3.343ex;" alt="{\displaystyle dl^{2}=\gamma _{\alpha \beta }\left(x^{1},x^{2},x^{3}\right)dx^{\alpha }dx^{\beta },}"></span></dd></dl> <p>then after transformation the same line element is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dl^{2}=\gamma _{\alpha \beta }\left(x^{\prime 1},x^{\prime 2},x^{\prime 3}\right)dx^{\prime \alpha }dx^{\prime \beta },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>α</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dl^{2}=\gamma _{\alpha \beta }\left(x^{\prime 1},x^{\prime 2},x^{\prime 3}\right)dx^{\prime \alpha }dx^{\prime \beta },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a730bd603a66f43874a3b4a1bb82a76f5b5c286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.074ex; height:3.343ex;" alt="{\displaystyle dl^{2}=\gamma _{\alpha \beta }\left(x^{\prime 1},x^{\prime 2},x^{\prime 3}\right)dx^{\prime \alpha }dx^{\prime \beta },}"></span></dd></dl> <p>with the same functional dependence of γ<sub>αβ</sub> on the new coordinates. (For a more theoretical and coordinate-independent definition of homogeneous space see <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous space</a>). A space is homogeneous if it admits a set of transformations (<a href="/wiki/Motion_(geometry)" title="Motion (geometry)"><i>a group of motions</i></a>) 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. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Frame_fields.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Frame_fields.svg/250px-Frame_fields.svg.png" decoding="async" width="250" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Frame_fields.svg/375px-Frame_fields.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Frame_fields.svg/500px-Frame_fields.svg.png 2x" data-file-width="460" data-file-height="434" /></a><figcaption><b>Figure 2.</b> The triad <i>e</i><sup>(<i>a</i>)</sup> (<i>e</i><sup>(1)</sup>, <i>e</i><sup>(2)</sup>, <i>e</i><sup>(3)</sup>) is an <a href="/wiki/Affine_coordinate_system" class="mw-redirect" title="Affine coordinate system">affine coordinate system</a> (including as a special case Cartesian coordinate system) whose coordinates are functions of the curvilinear coordinates x<sub>α</sub> (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>).</figcaption></figure> <p>In <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> the homogeneity of space is expressed by the <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariance</a> of the metric under parallel displacements (<a href="/wiki/Translation_(physics)" class="mw-redirect" title="Translation (physics)">translations</a>) of the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>. 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 (<i>dx</i>, <i>dy</i>, <i>dz</i>) 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 linear <a href="/wiki/Differential_forms" class="mw-redirect" title="Differential forms">differential forms</a>, which do not, however, reduce to <a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total differentials</a> of any coordinate functions. These forms are written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754c8bae65e0b3f9d9a558401b1c60cd3c6e388d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.294ex; height:3.343ex;" alt="{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}"></span> where the Latin index (<i>a</i>) labels three independent vectors (coordinate functions); these vectors are called a <a href="/wiki/Frame_field" class="mw-redirect" title="Frame field">frame field</a> or triad. The Greek letters label the three space-like <a href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear coordinates</a>. A spatial metric invariant is constructed under the given group of motions with the use of the above forms: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dl^{2}=\eta _{ab}\left(e_{\alpha }^{(a)}dx^{\alpha }\right)\left(e_{\beta }^{(b)}dx^{\beta }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dl^{2}=\eta _{ab}\left(e_{\alpha }^{(a)}dx^{\alpha }\right)\left(e_{\beta }^{(b)}dx^{\beta }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a952f7833dae34656af81888ef8ce728e78780cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.664ex; height:4.843ex;" alt="{\displaystyle dl^{2}=\eta _{ab}\left(e_{\alpha }^{(a)}dx^{\alpha }\right)\left(e_{\beta }^{(b)}dx^{\beta }\right)}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6a" class="reference nourlexpansion" style="font-weight:bold;">eq. 6a</span>)</b></td></tr></tbody></table> <p>i.e. the metric tensor is </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\alpha \beta }=\eta _{ab}e_{\alpha }^{(a)}e_{\beta }^{(b)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\alpha \beta }=\eta _{ab}e_{\alpha }^{(a)}e_{\beta }^{(b)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2febd8f4b7e6da4096dfc0e6bca03eb29f58077d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.257ex; height:4.009ex;" alt="{\displaystyle \gamma _{\alpha \beta }=\eta _{ab}e_{\alpha }^{(a)}e_{\beta }^{(b)}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6b" class="reference nourlexpansion" style="font-weight:bold;">eq. 6b</span>)</b></td></tr></tbody></table> <p>where the coefficients η<sub><i>ab</i></sub>, which are symmetric in the indices <i>a</i> and <i>b</i>, 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<sub><i>ab</i></sub> is not diagonal). </p><p>The reciprocal triple of vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(a)}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(a)}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ed6a6f168afadd73751591ea6b7b2505f6a594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.465ex; height:3.343ex;" alt="{\displaystyle e_{(a)}^{\alpha }}"></span> is introduced with the help of <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(a)}^{\alpha }e_{\alpha }^{(b)}=\delta _{a}^{b};\quad e_{(a)}^{\alpha }e_{\beta }^{(a)}=\delta _{\alpha }^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>;</mo> <mspace width="1em" /> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(a)}^{\alpha }e_{\alpha }^{(b)}=\delta _{a}^{b};\quad e_{(a)}^{\alpha }e_{\beta }^{(a)}=\delta _{\alpha }^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500a27454edbdad774862460d3b39f4288d37719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:27.699ex; height:4.176ex;" alt="{\displaystyle e_{(a)}^{\alpha }e_{\alpha }^{(b)}=\delta _{a}^{b};\quad e_{(a)}^{\alpha }e_{\beta }^{(a)}=\delta _{\alpha }^{\beta }}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6c" class="reference nourlexpansion" style="font-weight:bold;">eq. 6c</span>)</b></td></tr></tbody></table> <p>In the three-dimensional case, the relation between the two vector triples can be written explicitly </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{(1)}={\frac {1}{v}}\mathbf {e} ^{(2)}\times \mathbf {e} ^{(3)},\quad \mathbf {e} _{(2)}={\frac {1}{v}}\mathbf {e} ^{(3)}\times \mathbf {e} ^{(1)},\quad \mathbf {e} _{(3)}={\frac {1}{v}}\mathbf {e} ^{(1)}\times \mathbf {e} ^{(2)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{(1)}={\frac {1}{v}}\mathbf {e} ^{(2)}\times \mathbf {e} ^{(3)},\quad \mathbf {e} _{(2)}={\frac {1}{v}}\mathbf {e} ^{(3)}\times \mathbf {e} ^{(1)},\quad \mathbf {e} _{(3)}={\frac {1}{v}}\mathbf {e} ^{(1)}\times \mathbf {e} ^{(2)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/972bbcad678a9cf3ec3b4dbe952270e10defaa6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.201ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{(1)}={\frac {1}{v}}\mathbf {e} ^{(2)}\times \mathbf {e} ^{(3)},\quad \mathbf {e} _{(2)}={\frac {1}{v}}\mathbf {e} ^{(3)}\times \mathbf {e} ^{(1)},\quad \mathbf {e} _{(3)}={\frac {1}{v}}\mathbf {e} ^{(1)}\times \mathbf {e} ^{(2)},}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6d" class="reference nourlexpansion" style="font-weight:bold;">eq. 6d</span>)</b></td></tr></tbody></table> <p>where the volume <i>v</i> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\left\vert e_{\alpha }^{(a)}\right\vert =\mathbf {e} ^{(1)}\cdot \mathbf {e} ^{(2)}\times \mathbf {e} ^{(3)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow> <mo>|</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>|</mo> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\left\vert e_{\alpha }^{(a)}\right\vert =\mathbf {e} ^{(1)}\cdot \mathbf {e} ^{(2)}\times \mathbf {e} ^{(3)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08103202ddc06215012d87b130f1e1218f1d067d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:27.926ex; height:4.176ex;" alt="{\displaystyle v=\left\vert e_{\alpha }^{(a)}\right\vert =\mathbf {e} ^{(1)}\cdot \mathbf {e} ^{(2)}\times \mathbf {e} ^{(3)},}"></span></dd></dl> <p>with <b>e</b><sub>(<i>a</i>)</sub> and <b>e</b><sup>(<i>a</i>)</sup> regarded as Cartesian vectors with components <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(a)}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(a)}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ed6a6f168afadd73751591ea6b7b2505f6a594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.465ex; height:3.343ex;" alt="{\displaystyle e_{(a)}^{\alpha }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\alpha }^{(a)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\alpha }^{(a)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c853199a9c7a3433a224654104ca5e71b26889c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.465ex; height:3.343ex;" alt="{\displaystyle e_{\alpha }^{(a)}}"></span>, respectively. The <a href="/wiki/Determinant" title="Determinant">determinant</a> of the metric tensor <b><a href="#math_eq._6b">eq. 6b</a></b> is γ = η<i>v</i><sup>2</sup> where η is the determinant of the matrix η<sub><i>ab</i></sub>. </p><p>The required conditions for the homogeneity of the space are </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial e_{\alpha }^{(c)}}{\partial x^{\beta }}}-{\frac {\partial e_{\beta }^{(c)}}{\partial x^{\alpha }}}\right)e_{(a)}^{\alpha }e_{(b)}^{\beta }=C_{ab}^{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial e_{\alpha }^{(c)}}{\partial x^{\beta }}}-{\frac {\partial e_{\beta }^{(c)}}{\partial x^{\alpha }}}\right)e_{(a)}^{\alpha }e_{(b)}^{\beta }=C_{ab}^{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fa862dbcd1efc67ac078d22818e5d016befc79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:32.196ex; height:8.843ex;" alt="{\displaystyle \left({\frac {\partial e_{\alpha }^{(c)}}{\partial x^{\beta }}}-{\frac {\partial e_{\beta }^{(c)}}{\partial x^{\alpha }}}\right)e_{(a)}^{\alpha }e_{(b)}^{\beta }=C_{ab}^{c}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6e" class="reference nourlexpansion" style="font-weight:bold;">eq. 6e</span>)</b></td></tr></tbody></table> <p>The constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74defdb40853a227a7df13fe27c636dce3e2b189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.469ex; height:2.843ex;" alt="{\displaystyle C_{ab}^{c}}"></span> are called the <a href="/wiki/Structure_constants" title="Structure constants">structure constants</a> of the group. </p> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Proof of <b><a href="#math_eq._6e">eq. 6e</a></b> </th></tr> <tr> <td> <p>The invariance of the differential forms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754c8bae65e0b3f9d9a558401b1c60cd3c6e388d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.294ex; height:3.343ex;" alt="{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}"></span> means that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\alpha }^{(a)}(x)dx^{\alpha }=e_{\alpha }^{(a)}(x^{\prime })dx^{\prime \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\alpha }^{(a)}(x)dx^{\alpha }=e_{\alpha }^{(a)}(x^{\prime })dx^{\prime \alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a7e82c835b13a88e3e491123da72e7597f8dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.102ex; height:3.509ex;" alt="{\displaystyle e_{\alpha }^{(a)}(x)dx^{\alpha }=e_{\alpha }^{(a)}(x^{\prime })dx^{\prime \alpha }}"></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\alpha }^{(a)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\alpha }^{(a)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c853199a9c7a3433a224654104ca5e71b26889c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.465ex; height:3.343ex;" alt="{\displaystyle e_{\alpha }^{(a)}}"></span> on the two sides of the equation are the same functions of the old and new coordinates, respectively. Multiplying this equation by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(a)}^{\beta }(x^{\prime })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(a)}^{\beta }(x^{\prime })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4556b720faa5fe781b7a6a3f64c5e2705e6db6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.288ex; height:4.009ex;" alt="{\displaystyle e_{(a)}^{\beta }(x^{\prime })}"></span>, setting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{\prime \beta }={\frac {\partial x^{\prime \beta }}{\partial x^{\alpha }}}dx^{\alpha },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx^{\prime \beta }={\frac {\partial x^{\prime \beta }}{\partial x^{\alpha }}}dx^{\alpha },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc8be996a6db7a7e91844d8e0de2b6bd42bee4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.858ex; height:5.843ex;" alt="{\displaystyle dx^{\prime \beta }={\frac {\partial x^{\prime \beta }}{\partial x^{\alpha }}}dx^{\alpha },}"></span> and comparing coefficients of the same differentials <i>dx</i><sup>α</sup>, one finds </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial x^{\prime \beta }}{\partial x^{\alpha }}}=e_{(a)}^{\beta }(x^{\prime })e_{\alpha }^{(a)}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial x^{\prime \beta }}{\partial x^{\alpha }}}=e_{(a)}^{\beta }(x^{\prime })e_{\alpha }^{(a)}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16faca11d480f968f30adc969db6cce7b3763fda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.748ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial x^{\prime \beta }}{\partial x^{\alpha }}}=e_{(a)}^{\beta }(x^{\prime })e_{\alpha }^{(a)}(x).}"></span></dd></dl> <p>These equations are a system of differential equations that determine the functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime \beta }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\prime \beta }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11dc90daae685472258221506b40942b7b9cba49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.095ex; height:3.176ex;" alt="{\displaystyle x^{\prime \beta }(x)}"></span> for a given frame. In order to be integrable, these equations must satisfy identically the conditions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}x^{\prime \beta }}{\partial x^{\alpha }\partial x^{\gamma }}}={\frac {\partial ^{2}x^{\prime \beta }}{\partial x^{\gamma }\partial x^{\alpha }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}x^{\prime \beta }}{\partial x^{\alpha }\partial x^{\gamma }}}={\frac {\partial ^{2}x^{\prime \beta }}{\partial x^{\gamma }\partial x^{\alpha }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1372e8428a149778128d9e3edd420d2b1fdf7b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.827ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial ^{2}x^{\prime \beta }}{\partial x^{\alpha }\partial x^{\gamma }}}={\frac {\partial ^{2}x^{\prime \beta }}{\partial x^{\gamma }\partial x^{\alpha }}}.}"></span></dd></dl> <p>Calculating the derivatives, one finds </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {\partial e_{(a)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(b)}^{\delta }(x^{\prime })-{\frac {\partial e_{(b)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(a)}^{\delta }(x^{\prime })\right]e_{\gamma }^{(b)}(x)e_{\alpha }^{(a)}(x)=e_{(a)}^{\beta }(x^{\prime })\left[{\frac {\partial e_{\gamma }^{(a)}(x)}{\partial x^{\alpha }}}-{\frac {\partial e_{\alpha }^{(a)}(x)}{\partial x^{\gamma }}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>δ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>δ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\partial e_{(a)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(b)}^{\delta }(x^{\prime })-{\frac {\partial e_{(b)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(a)}^{\delta }(x^{\prime })\right]e_{\gamma }^{(b)}(x)e_{\alpha }^{(a)}(x)=e_{(a)}^{\beta }(x^{\prime })\left[{\frac {\partial e_{\gamma }^{(a)}(x)}{\partial x^{\alpha }}}-{\frac {\partial e_{\alpha }^{(a)}(x)}{\partial x^{\gamma }}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/624f67580e800b4397e489d050f1fdb3d8dbbad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:87.38ex; height:8.843ex;" alt="{\displaystyle \left[{\frac {\partial e_{(a)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(b)}^{\delta }(x^{\prime })-{\frac {\partial e_{(b)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(a)}^{\delta }(x^{\prime })\right]e_{\gamma }^{(b)}(x)e_{\alpha }^{(a)}(x)=e_{(a)}^{\beta }(x^{\prime })\left[{\frac {\partial e_{\gamma }^{(a)}(x)}{\partial x^{\alpha }}}-{\frac {\partial e_{\alpha }^{(a)}(x)}{\partial x^{\gamma }}}\right].}"></span></dd></dl> <p>Multiplying both sides of the equations by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(d)}^{\alpha }(x)e_{(c)}^{\gamma }(x)e_{\beta }^{(f)}(x^{\prime })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(d)}^{\alpha }(x)e_{(c)}^{\gamma }(x)e_{\beta }^{(f)}(x^{\prime })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6acb0057d499182b8fd5ac3530cd378ec5e3b3cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:20.363ex; height:4.176ex;" alt="{\displaystyle e_{(d)}^{\alpha }(x)e_{(c)}^{\gamma }(x)e_{\beta }^{(f)}(x^{\prime })}"></span> and shifting the differentiation from one factor to the other by using <b><a href="#math_eq._6c">eq. 6c</a></b>, one gets for the left side: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\beta }^{(f)}(x^{\prime })\left[{\frac {\partial e_{(d)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(c)}^{\delta }(x^{\prime })-{\frac {\partial e_{(c)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(d)}^{\delta }(x^{\prime })\right]=e_{(c)}^{\beta }(x^{\prime })e_{(d)}^{\delta }(x^{\prime })\left[{\frac {\partial e_{\beta }^{(f)}(x^{\prime })}{\partial x^{\prime \delta }}}-{\frac {\partial e_{\delta }^{(f)}(x^{\prime })}{\partial x^{\prime \beta }}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>δ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>δ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>δ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi>β</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\beta }^{(f)}(x^{\prime })\left[{\frac {\partial e_{(d)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(c)}^{\delta }(x^{\prime })-{\frac {\partial e_{(c)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(d)}^{\delta }(x^{\prime })\right]=e_{(c)}^{\beta }(x^{\prime })e_{(d)}^{\delta }(x^{\prime })\left[{\frac {\partial e_{\beta }^{(f)}(x^{\prime })}{\partial x^{\prime \delta }}}-{\frac {\partial e_{\delta }^{(f)}(x^{\prime })}{\partial x^{\prime \beta }}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75343498a283decbda03092f935254bc02347d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:90.602ex; height:8.843ex;" alt="{\displaystyle e_{\beta }^{(f)}(x^{\prime })\left[{\frac {\partial e_{(d)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(c)}^{\delta }(x^{\prime })-{\frac {\partial e_{(c)}^{\beta }(x^{\prime })}{\partial x^{\prime \delta }}}e_{(d)}^{\delta }(x^{\prime })\right]=e_{(c)}^{\beta }(x^{\prime })e_{(d)}^{\delta }(x^{\prime })\left[{\frac {\partial e_{\beta }^{(f)}(x^{\prime })}{\partial x^{\prime \delta }}}-{\frac {\partial e_{\delta }^{(f)}(x^{\prime })}{\partial x^{\prime \beta }}}\right].}"></span></dd></dl> <p>and for the right, the same expression in the variable <i>x</i>. Since <i>x</i> and <i>x'</i> are arbitrary, these expression must reduce to constants to obtain <b><a href="#math_eq._6e">eq. 6e</a></b>. </p> </td></tr></tbody></table></dd></dl> <p>Multiplying by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(c)}^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(c)}^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d627323a133281798348f905caa11c0a1e81daeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.307ex; height:3.676ex;" alt="{\displaystyle e_{(c)}^{\gamma }}"></span>, <b><a href="#math_eq._6e">eq. 6e</a></b> can be rewritten in the form </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(a)}^{\alpha }{\frac {\partial e_{(b)}^{\gamma }}{\partial x^{\alpha }}}-e_{(b)}^{\beta }{\frac {\partial e_{(a)}^{\gamma }}{\partial x^{\beta }}}=C_{ab}^{c}e_{(c)}^{\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(a)}^{\alpha }{\frac {\partial e_{(b)}^{\gamma }}{\partial x^{\alpha }}}-e_{(b)}^{\beta }{\frac {\partial e_{(a)}^{\gamma }}{\partial x^{\beta }}}=C_{ab}^{c}e_{(c)}^{\gamma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8cc7b951945828ce78e555b3e327f23730244d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.201ex; height:6.843ex;" alt="{\displaystyle e_{(a)}^{\alpha }{\frac {\partial e_{(b)}^{\gamma }}{\partial x^{\alpha }}}-e_{(b)}^{\beta }{\frac {\partial e_{(a)}^{\gamma }}{\partial x^{\beta }}}=C_{ab}^{c}e_{(c)}^{\gamma }.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6f" class="reference nourlexpansion" style="font-weight:bold;">eq. 6f</span>)</b></td></tr></tbody></table> <p><b><a href="#math_Equation_6e">Equation 6e</a></b> can be written in a vector form as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{(a)}\times \mathbf {e} _{(b)}{\text{curl }}\mathbf {e} ^{(c)}=-C_{ab}^{c},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>curl </mtext> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{(a)}\times \mathbf {e} _{(b)}{\text{curl }}\mathbf {e} ^{(c)}=-C_{ab}^{c},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b795255b30f947779de92808d20de27a2909c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.824ex; height:3.676ex;" alt="{\displaystyle \mathbf {e} _{(a)}\times \mathbf {e} _{(b)}{\text{curl }}\mathbf {e} ^{(c)}=-C_{ab}^{c},}"></span></dd></dl> <p>where again the vector operations are done as if the coordinates <i>x</i><sup>α</sup> were Cartesian. Using <b><a href="#math_eq._6d">eq. 6d</a></b>, one obtains </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{v}}\left(\mathbf {e} ^{(1)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{32}^{1},\quad {\frac {1}{v}}\left(\mathbf {e} ^{(2)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{13}^{1},\quad {\frac {1}{v}}\left(\mathbf {e} ^{(3)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{21}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>curl </mtext> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>curl </mtext> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>curl </mtext> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{v}}\left(\mathbf {e} ^{(1)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{32}^{1},\quad {\frac {1}{v}}\left(\mathbf {e} ^{(2)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{13}^{1},\quad {\frac {1}{v}}\left(\mathbf {e} ^{(3)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{21}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570eff4ccf890aa8ab3fbf18c7ae7b4e0209f89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:76.851ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{v}}\left(\mathbf {e} ^{(1)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{32}^{1},\quad {\frac {1}{v}}\left(\mathbf {e} ^{(2)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{13}^{1},\quad {\frac {1}{v}}\left(\mathbf {e} ^{(3)}{\text{curl }}\mathbf {e} ^{(1)}\right)=C_{21}^{1}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6g" class="reference nourlexpansion" style="font-weight:bold;">eq. 6g</span>)</b></td></tr></tbody></table> <p>and six more equations obtained by a cyclic permutation of indices 1, 2, 3. </p><p>The structure constants are antisymmetric in their lower indices as seen from their definition <b><a href="#math_eq._6e">eq. 6e</a></b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}=-C_{ba}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}=-C_{ba}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a74adf5ba3375b3894ce6865a4347049bc28c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.845ex; height:2.843ex;" alt="{\displaystyle C_{ab}^{c}=-C_{ba}^{c}}"></span>. Another condition on the structure constants can be obtained by noting that <b><a href="#math_eq._6f">eq. 6f</a></b> can be written in the form of <a href="/wiki/Commutator" title="Commutator">commutation relations</a> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[X_{a},X_{b}\right]\equiv X_{a}X_{b}-X_{b}X_{a}=C_{ab}^{c}X_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>≡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[X_{a},X_{b}\right]\equiv X_{a}X_{b}-X_{b}X_{a}=C_{ab}^{c}X_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe8fb3e8dae606acce90cc9a9902fb4d7b2492b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.367ex; height:3.009ex;" alt="{\displaystyle \left[X_{a},X_{b}\right]\equiv X_{a}X_{b}-X_{b}X_{a}=C_{ab}^{c}X_{c}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6h" class="reference nourlexpansion" style="font-weight:bold;">eq. 6h</span>)</b></td></tr></tbody></table> <p>for the linear <a href="/wiki/Differential_operators" class="mw-redirect" title="Differential operators">differential operators</a> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{a}=e_{(a)}^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{a}=e_{(a)}^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d55236b994242b792e7c66597dc0f6b93dffb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.357ex; height:5.509ex;" alt="{\displaystyle X_{a}=e_{(a)}^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6i" class="reference nourlexpansion" style="font-weight:bold;">eq. 6i</span>)</b></td></tr></tbody></table> <p>In the mathematical theory of continuous groups (<a href="/wiki/Lie_groups" class="mw-redirect" title="Lie groups">Lie groups</a>) the operators <i>X</i><sub><i>a</i></sub> satisfying conditions <b><a href="#math_eq._6h">eq. 6h</a></b> are called the <a href="/wiki/Generator_(groups)" class="mw-redirect" title="Generator (groups)">generators of the group</a>. The theory of Lie groups uses operators defined using the <a href="/wiki/Killing_vectors" class="mw-redirect" title="Killing vectors">Killing vectors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{(a)}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{(a)}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974347e2faa83992c1bc9485ec9037d0ee6a93db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.4ex; height:3.343ex;" alt="{\displaystyle \xi _{(a)}^{\alpha }}"></span> instead of triads <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(a)}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{(a)}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ed6a6f168afadd73751591ea6b7b2505f6a594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.465ex; height:3.343ex;" alt="{\displaystyle e_{(a)}^{\alpha }}"></span>. Since in the synchronous metric none of the γ<sub>αβ</sub> components depends on time, the Killing vectors (triads) are time-like. </p><p>The conditions <b><a href="#math_eq._6h">eq. 6h</a></b> follow from the <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [[X_{a},X_{b}],X_{c}]+[[X_{b},X_{c}],X_{a}]+[[X_{c},X_{a}],X_{b}]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [[X_{a},X_{b}],X_{c}]+[[X_{b},X_{c}],X_{a}]+[[X_{c},X_{a}],X_{b}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b257669da84cabe273a083e469cdb1accdb50d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.177ex; height:2.843ex;" alt="{\displaystyle [[X_{a},X_{b}],X_{c}]+[[X_{b},X_{c}],X_{a}]+[[X_{c},X_{a}],X_{b}]=0}"></span></dd></dl> <p>and have the form </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{e}C_{ec}^{d}+C_{bc}^{e}C_{ea}^{d}+C_{ca}^{e}C_{eb}^{d}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{e}C_{ec}^{d}+C_{bc}^{e}C_{ea}^{d}+C_{ca}^{e}C_{eb}^{d}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45420a45a0aaba97261dcf68d3e29654fe7d441c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.465ex; height:3.176ex;" alt="{\displaystyle C_{ab}^{e}C_{ec}^{d}+C_{bc}^{e}C_{ea}^{d}+C_{ca}^{e}C_{eb}^{d}=0}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6j" class="reference nourlexpansion" style="font-weight:bold;">eq. 6j</span>)</b></td></tr></tbody></table> <p>It is a definite advantage to use, in place of the three-index constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74defdb40853a227a7df13fe27c636dce3e2b189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.469ex; height:2.843ex;" alt="{\displaystyle C_{ab}^{c}}"></span>, a set of two-index quantities, obtained by the dual transformation </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ab}^{c}=e_{abd}C^{dc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>d</mi> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ab}^{c}=e_{abd}C^{dc}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd0c71a98533d09e16c4089649b3746ca6a1801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.92ex; height:3.343ex;" alt="{\displaystyle C_{ab}^{c}=e_{abd}C^{dc}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6k" class="reference nourlexpansion" style="font-weight:bold;">eq. 6k</span>)</b></td></tr></tbody></table> <p>where <i>e<sub>abc</sub></i> = <i>e<sup>abc</sup></i> is the <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">unit antisymmetric symbol</a> (with <i>e</i><sub>123</sub> = +1). With these constants the commutation relations <b><a href="#math_eq._6h">eq. 6h</a></b> are written as </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{abc}X_{b}X_{c}=C^{ad}X_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>d</mi> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{abc}X_{b}X_{c}=C^{ad}X_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04450fc85718b3e08143c94123b4ad0e213ae0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.207ex; height:3.009ex;" alt="{\displaystyle e^{abc}X_{b}X_{c}=C^{ad}X_{d}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6l" class="reference nourlexpansion" style="font-weight:bold;">eq. 6l</span>)</b></td></tr></tbody></table> <p>The antisymmetry property is already taken into account in the definition <b><a href="#math_eq._6k">eq. 6k</a></b>, while property <b><a href="#math_eq._6j">eq. 6j</a></b> takes the form </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{bcd}C^{cd}C^{ba}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{bcd}C^{cd}C^{ba}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64e841f64ad9b391aedf8213752edfe04d01840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.061ex; height:3.009ex;" alt="{\displaystyle e_{bcd}C^{cd}C^{ba}=0}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6m" class="reference nourlexpansion" style="font-weight:bold;">eq. 6m</span>)</b></td></tr></tbody></table> <p>The choice of the three frame vectors in the differential forms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754c8bae65e0b3f9d9a558401b1c60cd3c6e388d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.294ex; height:3.343ex;" alt="{\displaystyle e_{\alpha }^{(a)}dx^{\alpha }}"></span> (and with them the operators <i>X<sub>a</sub></i>) is not unique. They can be subjected to any linear transformation with constant coefficients: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{(a)}=A_{a}^{b}\mathbf {e} _{(b)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{(a)}=A_{a}^{b}\mathbf {e} _{(b)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80766bacd12a20b0f1e4cac704ce4d176f73d460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.639ex; height:3.343ex;" alt="{\displaystyle \mathbf {e} _{(a)}=A_{a}^{b}\mathbf {e} _{(b)}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6n" class="reference nourlexpansion" style="font-weight:bold;">eq. 6n</span>)</b></td></tr></tbody></table> <p>The quantities η<sub><i>ab</i></sub> and <i>C<sup>ab</sup></i> behave like tensors (are invariant) with respect to such transformations. </p><p>The conditions <b><a href="#math_eq._6m">eq. 6m</a></b> 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 type <b><a href="#math_eq._6n">eq. 6n</a></b>. 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 quantities <i>C<sup>ab</sup></i>, by the following simple method (C. G. Behr, 1962). </p><p>The asymmetric tensor <i>C<sup>ab</sup></i> can be resolved into a symmetric and an antisymmetric part. The first is denoted by <i>n<sup>ab</sup></i>, and the second is expressed in terms of its <a href="/wiki/Dual_vector" class="mw-redirect" title="Dual vector">dual vector</a> <i>a<sub>c</sub></i>: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{ab}=n^{ab}+e^{abc}a_{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{ab}=n^{ab}+e^{abc}a_{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec0ac398d87d68b0dbeb1e6752f0a234013cf38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.169ex; height:3.009ex;" alt="{\displaystyle C^{ab}=n^{ab}+e^{abc}a_{c}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6o" class="reference nourlexpansion" style="font-weight:bold;">eq. 6o</span>)</b></td></tr></tbody></table> <p>Substitution of this expression in <b><a href="#math_eq._6m">eq. 6m</a></b> leads to the condition </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{ab}a_{b}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{ab}a_{b}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e572b936a032b85b57013482c4e38d0c7b3b6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.277ex; height:3.009ex;" alt="{\displaystyle n^{ab}a_{b}=0.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6p" class="reference nourlexpansion" style="font-weight:bold;">eq. 6p</span>)</b></td></tr></tbody></table> <p>By means of the transformations <b><a href="#math_eq._6n">eq. 6n</a></b> the symmetric tensor <i>n<sup>ab</sup></i> can be brought to diagonal form with <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub>. Equation <b><a href="#math_6p">6p</a></b> shows that the vector <i>a<sub>b</sub></i> (if it exists) lies along one of the principal directions of the tensor <i>n<sup>ab</sup></i>, the one corresponding to the eigenvalue zero. Without loss of generality one can therefore set <i>a<sub>b</sub></i> = (<i>a</i>, 0, 0). Then <b><a href="#math_eq._6p">eq. 6p</a></b> reduces to <i>an</i><sub>1</sub> = 0, i.e. one of the quantities <i>a</i> or <i>n</i><sub>1</sub> must be zero. The Jacobi identities take the form: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X_{1},X_{2}]=-aX_{2}+n_{3}X_{3},\quad [X_{2},X_{3}]=n_{1}X_{1},\quad [X_{3},X_{1}]=n_{2}X_{2}+aX_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <mi>a</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X_{1},X_{2}]=-aX_{2}+n_{3}X_{3},\quad [X_{2},X_{3}]=n_{1}X_{1},\quad [X_{3},X_{1}]=n_{2}X_{2}+aX_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4be2d14a88dbd4628d84fba5ab875ed708c2593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.697ex; height:2.843ex;" alt="{\displaystyle [X_{1},X_{2}]=-aX_{2}+n_{3}X_{3},\quad [X_{2},X_{3}]=n_{1}X_{1},\quad [X_{3},X_{1}]=n_{2}X_{2}+aX_{3}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._6q" class="reference nourlexpansion" style="font-weight:bold;">eq. 6q</span>)</b></td></tr></tbody></table> <p>The only remaining freedoms are sign changes of the operators <i>X<sub>a</sub></i> and their multiplication by arbitrary constants. This permits to simultaneously change the sign of all the <i>n<sub>a</sub></i> and also to make the quantity <i>a</i> positive (if it is different from zero). Also all structure constants can be made equal to ±1, if at least one of the quantities <i>a</i>, <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub> vanishes. But if all three of these quantities differ from zero, the scale transformations leave invariant the ratio <i>h</i> = <i>a</i><sup>2</sup>(<i>n</i><sub>2</sub><i>n</i><sub>3</sub>)<sup>−1</sup>. </p><p>Thus one arrives at the Bianchi classification listing the possible types of homogeneous spaces classified by the values of <i>a</i>, <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, <i>n</i><sub>3</sub> which is graphically presented in Fig. 3. In the class A case (<i>a</i> = 0), <b>type IX</b> (<i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=1, <i>n</i><sup>(3)</sup>=1) is represented by octant 2, <b>type VIII</b> (<i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=1, <i>n</i><sup>(3)</sup>=–1) is represented by octant 6, while <b>type VII<sub>0</sub></b> (<i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=1, <i>n</i><sup>(3)</sup>=0) is represented by the first quadrant of the horizontal plane and <b>type VI<sub>0</sub></b> (<i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=–1, <i>n</i><sup>(3)</sup>=0) is represented by the fourth quadrant of this plane; <b>type II</b> ((<i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=0, <i>n</i><sup>(3)</sup>=0) is represented by the interval [0,1] along <i>n</i><sup>(1)</sup> and <b>type I</b> (<i>n</i><sup>(1)</sup>=0, <i>n</i><sup>(2)</sup>=0, <i>n</i><sup>(3)</sup>=0) is at the origin. Similarly in the class B case (with <i>n</i><sup>(3)</sup> = 0), Bianchi <b>type VI<sub>h</sub></b> (<i>a</i>=<i>h</i>, <i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=–1) projects to the fourth quadrant of the horizontal plane and <b>type VII<sub>h</sub></b> (<i>a</i>=<i>h</i>, <i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=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 function <i>h</i> = <i>a</i><sup>2</sup>(<i>n</i><sup>(1)</sup><i>n</i><sup>(2)</sup>)<sup>−1</sup>. A typical such surface is illustrated in one octant, the angle <i>θ</i> given by tan <i>θ</i> = |<i>h</i>/2|<sup>1/2</sup>; those in the remaining octants are obtained by rotation through multiples of <i>π</i>/2, <i>h</i> alternating in sign for a given magnitude |<i>h</i>|. <b>Type III</b> is a subtype of VI<sub>h</sub> with <i>a</i>=1. <b>Type V</b> (<i>a</i>=1, <i>n</i><sup>(1)</sup>=0, <i>n</i><sup>(2)</sup>=0) is the interval (0,1] along the axis <i>a</i> and <b>type IV</b> (<i>a</i>=1, <i>n</i><sup>(1)</sup>=1, <i>n</i><sup>(2)</sup>=0) is the vertical open face between the first and fourth quadrants of the <i>a</i> = 0 plane with the latter giving the class A limit of each type. </p><p>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: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{(a)(b)}=R_{\alpha \beta }e_{(a)}^{\alpha }e_{(b)}^{\beta },\quad R_{0(a)}=R_{0\alpha }e_{(a)}^{\alpha },\quad u^{(a)}=u^{\alpha }e_{\alpha }^{(a)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>,</mo> <mspace width="1em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>α</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>,</mo> <mspace width="1em" /> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{(a)(b)}=R_{\alpha \beta }e_{(a)}^{\alpha }e_{(b)}^{\beta },\quad R_{0(a)}=R_{0\alpha }e_{(a)}^{\alpha },\quad u^{(a)}=u^{\alpha }e_{\alpha }^{(a)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87038213b727a5516375fdc6bae92ac0081eb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:55.632ex; height:4.176ex;" alt="{\displaystyle R_{(a)(b)}=R_{\alpha \beta }e_{(a)}^{\alpha }e_{(b)}^{\beta },\quad R_{0(a)}=R_{0\alpha }e_{(a)}^{\alpha },\quad u^{(a)}=u^{\alpha }e_{\alpha }^{(a)},}"></span> </p><p>where all these quantities are now functions of <i>t</i> alone; the scalar quantities, the energy density ε and the pressure of the matter <i>p</i>, are also functions of the time. </p><p>The Einstein equations in vacuum in synchronous reference frame are<sup id="cite_ref-LK_11-0" class="reference"><a href="#cite_note-LK-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-LL_12-0" class="reference"><a href="#cite_note-LL-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-convention_13-0" class="reference"><a href="#cite_note-convention-13"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}^{0}=-{\frac {1}{2}}{\frac {\partial \varkappa _{\alpha }^{\alpha }}{\partial t}}-{\frac {1}{4}}\varkappa _{\alpha }^{\beta }\varkappa _{\beta }^{\alpha }=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}^{0}=-{\frac {1}{2}}{\frac {\partial \varkappa _{\alpha }^{\alpha }}{\partial t}}-{\frac {1}{4}}\varkappa _{\alpha }^{\beta }\varkappa _{\beta }^{\alpha }=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d866eb8ec6f7a7e8c66ff2635ca92348b90c497b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.625ex; height:5.509ex;" alt="{\displaystyle R_{0}^{0}=-{\frac {1}{2}}{\frac {\partial \varkappa _{\alpha }^{\alpha }}{\partial t}}-{\frac {1}{4}}\varkappa _{\alpha }^{\beta }\varkappa _{\beta }^{\alpha }=0,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._11" class="reference nourlexpansion" style="font-weight:bold;">eq. 11</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\alpha }^{\beta }=-{\frac {1}{2{\sqrt {-g}}}}{\frac {\partial }{\partial t}}\left({\sqrt {-g}}\varkappa _{\alpha }^{\beta }\right)-P_{\alpha }^{\beta }=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>g</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>g</mi> </msqrt> </mrow> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\alpha }^{\beta }=-{\frac {1}{2{\sqrt {-g}}}}{\frac {\partial }{\partial t}}\left({\sqrt {-g}}\varkappa _{\alpha }^{\beta }\right)-P_{\alpha }^{\beta }=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55760f85890e9621e21daf268646637959e523fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.888ex; height:6.343ex;" alt="{\displaystyle R_{\alpha }^{\beta }=-{\frac {1}{2{\sqrt {-g}}}}{\frac {\partial }{\partial t}}\left({\sqrt {-g}}\varkappa _{\alpha }^{\beta }\right)-P_{\alpha }^{\beta }=0,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._12" class="reference nourlexpansion" style="font-weight:bold;">eq. 12</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\alpha }^{0}={\frac {1}{2}}\left(\varkappa _{\alpha ;\beta }^{\beta }-\varkappa _{\beta ;\alpha }^{\beta }\right)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>;</mo> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>;</mo> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\alpha }^{0}={\frac {1}{2}}\left(\varkappa _{\alpha ;\beta }^{\beta }-\varkappa _{\beta ;\alpha }^{\beta }\right)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80acc8bbd42f8baeb3414ccac807db0a7fae719b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.523ex; height:5.176ex;" alt="{\displaystyle R_{\alpha }^{0}={\frac {1}{2}}\left(\varkappa _{\alpha ;\beta }^{\beta }-\varkappa _{\beta ;\alpha }^{\beta }\right)=0,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._13" class="reference nourlexpansion" style="font-weight:bold;">eq. 13</span>)</b></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varkappa _{\alpha }^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varkappa _{\alpha }^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23118dc98d4242dfa82f8d0ac055809eee57348f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.927ex; height:3.176ex;" alt="{\displaystyle \varkappa _{\alpha }^{\beta }}"></span> is the 3-dimensional tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varkappa _{\alpha }^{\beta }={\frac {\partial \gamma _{\alpha }^{\beta }}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varkappa _{\alpha }^{\beta }={\frac {\partial \gamma _{\alpha }^{\beta }}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1394771ca798e988e91e07d82ebf7e39691ff5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.668ex; height:6.176ex;" alt="{\displaystyle \varkappa _{\alpha }^{\beta }={\frac {\partial \gamma _{\alpha }^{\beta }}{\partial t}}}"></span>, and <i>P</i><sub>αβ</sub> is the 3-dimensional <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a>, which is expressed by the 3-dimensional <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric tensor</a> γ<sub>αβ</sub> in the same way as <i>R<sub>ik</sub></i> is expressed by <i>g<sub>ik</sub></i>; <i>P</i><sub>αβ</sub> contains only the space (but not the time) derivatives of γ<sub>αβ</sub>. Using triads, for <b><a href="#math_eq._11">eq. 11</a></b> one has simply </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varkappa _{(a)(b)}={\frac {\partial \eta _{ab}}{\partial t}},\quad \varkappa _{(a)}^{(b)}={\frac {\partial \eta _{ac}}{\partial t}}\eta ^{cb}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varkappa _{(a)(b)}={\frac {\partial \eta _{ab}}{\partial t}},\quad \varkappa _{(a)}^{(b)}={\frac {\partial \eta _{ac}}{\partial t}}\eta ^{cb}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75125e6aa7ecf272c5f19ffc5198e305c7f90c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.15ex; height:5.676ex;" alt="{\displaystyle \varkappa _{(a)(b)}={\frac {\partial \eta _{ab}}{\partial t}},\quad \varkappa _{(a)}^{(b)}={\frac {\partial \eta _{ac}}{\partial t}}\eta ^{cb}.}"></span></dd></dl> <p>The components of <i>P</i><sub>(<i>a</i>)(<i>b</i>)</sub> can be expressed in terms of the quantities η<sub><i>ab</i></sub> and the structure constants of the group by using the tetrad representation of the Ricci tensor in terms of quantities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{abc}=\left(e_{(a)i,k}-e_{(a)k,i}\right)e_{(b)}^{i}e_{(c)}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>k</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{abc}=\left(e_{(a)i,k}-e_{(a)k,i}\right)e_{(b)}^{i}e_{(c)}^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ee84b0c197f1bb1739724af043557c84376854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:29.63ex; height:3.676ex;" alt="{\displaystyle \lambda _{abc}=\left(e_{(a)i,k}-e_{(a)k,i}\right)e_{(b)}^{i}e_{(c)}^{k}}"></span><sup id="cite_ref-FOOTNOTELandauLifshitz1988eq._(98.14)_14-0" class="reference"><a href="#cite_note-FOOTNOTELandauLifshitz1988eq._(98.14)-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> <mo>,</mo> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> <mo>,</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msubsup> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msubsup> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msubsup> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436a8ca83dd621cfbf1ed2b9739fd07bc65655c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:94.134ex; height:6.176ex;" alt="{\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).}"></span></dd></dl> <p>After replacing the three-index symbols <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{bc}^{a}=C_{bc}^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{bc}^{a}=C_{bc}^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca36c75bdd751b95a26b36c705c5944169783843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.415ex; height:2.843ex;" alt="{\displaystyle \lambda _{bc}^{a}=C_{bc}^{a}}"></span> by two-index symbols <i>C</i><sup><i>ab</i></sup> and the transformations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{ad}\eta _{bc}\eta _{cf}e^{def}=\eta e_{abc},\quad e_{abf}e^{cdf}=\delta _{a}^{c}\delta _{b}^{d}-\delta _{a}^{d}\delta _{b}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>d</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>f</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msup> <mo>=</mo> <mi>η</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>f</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> <mi>f</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{ad}\eta _{bc}\eta _{cf}e^{def}=\eta e_{abc},\quad e_{abf}e^{cdf}=\delta _{a}^{c}\delta _{b}^{d}-\delta _{a}^{d}\delta _{b}^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96994dfb396fe338b6b3653c6daba9005caf099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.947ex; height:3.343ex;" alt="{\displaystyle \eta _{ad}\eta _{bc}\eta _{cf}e^{def}=\eta e_{abc},\quad e_{abf}e^{cdf}=\delta _{a}^{c}\delta _{b}^{d}-\delta _{a}^{d}\delta _{b}^{c}}"></span></dd></dl> <p>one gets the "homogeneous" Ricci tensor expressed in structure constants: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>2</mn> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>d</mi> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>b</mi> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>d</mi> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>a</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msup> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f425911c70fa1fb469b49955c1580fe049d668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:85.026ex; height:5.676ex;" alt="{\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\}.}"></span></dd></dl> <p>Here, all indices are raised and lowered with the local metric tensor η<sub><i>ab</i></sub> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{a}^{b}=\eta _{ac}C^{cb},\quad C_{ab}=\eta _{ac}\eta _{bd}C^{cd}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>d</mi> </mrow> </msub> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{a}^{b}=\eta _{ac}C^{cb},\quad C_{ab}=\eta _{ac}\eta _{bd}C^{cd}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59508d41a0b511fda4335efbab569c9cd637de15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.374ex; height:3.176ex;" alt="{\displaystyle C_{a}^{b}=\eta _{ac}C^{cb},\quad C_{ab}=\eta _{ac}\eta _{bd}C^{cd}.}"></span></dd></dl> <p>The <a href="/wiki/Bianchi_identities" class="mw-redirect" title="Bianchi identities">Bianchi identities</a> for the three-dimensional tensor <i>P</i><sub>αβ</sub> in the homogeneous space take the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{b}^{c}C_{ca}^{b}+P_{a}^{c}C_{cb}^{b}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{b}^{c}C_{ca}^{b}+P_{a}^{c}C_{cb}^{b}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a01825e580eaa069e094319258b28b7e9449c3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.067ex; height:3.176ex;" alt="{\displaystyle P_{b}^{c}C_{ca}^{b}+P_{a}^{c}C_{cb}^{b}=0.}"></span></dd></dl> <p>Taking into account the transformations of covariant derivatives for arbitrary four-vectors <i>A</i><sub><i>i</i></sub> and four-tensors <i>A</i><sub><i>ik</i></sub> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i;k}e_{(a)}^{i}e_{(b)}^{k}=A_{(a)(b)}-A^{(d)}\gamma _{dab},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>;</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i;k}e_{(a)}^{i}e_{(b)}^{k}=A_{(a)(b)}-A^{(d)}\gamma _{dab},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9200e44ee76403967d371fb08ee9ddb3b269c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:31.302ex; height:4.009ex;" alt="{\displaystyle A_{i;k}e_{(a)}^{i}e_{(b)}^{k}=A_{(a)(b)}-A^{(d)}\gamma _{dab},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mo>;</mo> <mi>l</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>−</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>a</mi> <mi>c</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23513bfad195d5c221c1496643391c892153c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:47.776ex; height:4.176ex;" alt="{\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},}"></span></dd></dl> <p>the final expressions for the triad components of the Ricci four-tensor are: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}^{0}=-{\frac {1}{2}}{\frac {\partial \varkappa _{(a)}^{(a)}}{\partial t}}-{\frac {1}{4}}\varkappa _{(a)}^{(b)}\varkappa _{(b)}^{(a)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}^{0}=-{\frac {1}{2}}{\frac {\partial \varkappa _{(a)}^{(a)}}{\partial t}}-{\frac {1}{4}}\varkappa _{(a)}^{(b)}\varkappa _{(b)}^{(a)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded7b6d664d70fef16565d86f1315369622128e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.601ex; height:7.176ex;" alt="{\displaystyle R_{0}^{0}=-{\frac {1}{2}}{\frac {\partial \varkappa _{(a)}^{(a)}}{\partial t}}-{\frac {1}{4}}\varkappa _{(a)}^{(b)}\varkappa _{(b)}^{(a)},}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._11a" class="reference nourlexpansion" style="font-weight:bold;">eq. 11a</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{(a)}^{(b)}=-{\frac {1}{2{\sqrt {\eta }}}}{\frac {\partial }{\partial t}}\left({\sqrt {\eta }}\varkappa _{(a)}^{(b)}\right)-P_{(a)}^{(b)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>η</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>η</mi> </msqrt> </mrow> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{(a)}^{(b)}=-{\frac {1}{2{\sqrt {\eta }}}}{\frac {\partial }{\partial t}}\left({\sqrt {\eta }}\varkappa _{(a)}^{(b)}\right)-P_{(a)}^{(b)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3790c72c09db04672f9dc61844104fbb70d2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.913ex; height:6.343ex;" alt="{\displaystyle R_{(a)}^{(b)}=-{\frac {1}{2{\sqrt {\eta }}}}{\frac {\partial }{\partial t}}\left({\sqrt {\eta }}\varkappa _{(a)}^{(b)}\right)-P_{(a)}^{(b)},}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._12a" class="reference nourlexpansion" style="font-weight:bold;">eq. 12a</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{(a)}^{0}=-{\frac {1}{2}}\varkappa _{(b)}^{(c)}\left(C_{ca}^{b}-\delta _{a}^{b}C_{dc}^{d}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>ϰ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{(a)}^{0}=-{\frac {1}{2}}\varkappa _{(b)}^{(c)}\left(C_{ca}^{b}-\delta _{a}^{b}C_{dc}^{d}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/920241933e1914b2bf2bc8239bc9b5e4e65f3fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.494ex; height:5.176ex;" alt="{\displaystyle R_{(a)}^{0}=-{\frac {1}{2}}\varkappa _{(b)}^{(c)}\left(C_{ca}^{b}-\delta _{a}^{b}C_{dc}^{d}\right).}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_eq._13a" class="reference nourlexpansion" style="font-weight:bold;">eq. 13a</span>)</b></td></tr></tbody></table> <p>In setting up the Einstein equations there is thus no need to use explicit expressions for the basis vectors as functions of the coordinates. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/25px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/37px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/49px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li></ul> <ul><li><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Table of Lie groups</a></li> <li><a href="/wiki/List_of_simple_Lie_groups" class="mw-redirect" title="List of simple Lie groups">List of simple Lie groups</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-convention-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-convention_13-0">^</a></b></span> <span class="reference-text">The convention used by BKL is the same as in the <a href="#CITEREFLandauLifshitz1988">Landau & Lifshitz (1988)</a> book. The Latin indices run through the values 0, 1, 2, 3; Greek indices run through the space values 1, 2, 3. The metric <i>g<sub>ik</sub></i> has the signature (+ − − −); γ<sub>αβ</sub> = −<i>g</i><sub>αβ</sub> is the 3-dimensional space metric tensor. BKL use a system of units, in which the speed of light and the Einstein gravitational constant are equal to 1.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTELandauLifshitz1988-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELandauLifshitz1988_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLandauLifshitz1988">Landau & Lifshitz 1988</a>.</span> </li> <li id="cite_note-FOOTNOTEWald1984-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWald1984_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWald1984">Wald 1984</a>.</span> </li> <li id="cite_note-FOOTNOTEBelinskyKhalatnikovLifshitz1971-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBelinskyKhalatnikovLifshitz1971_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBelinskyKhalatnikovLifshitz1971">Belinsky, Khalatnikov & Lifshitz 1971</a>.</span> </li> <li id="cite_note-FOOTNOTEBelinskyKhalatnikovLifshitz1972-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBelinskyKhalatnikovLifshitz1972_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBelinskyKhalatnikovLifshitz1972">Belinsky, Khalatnikov & Lifshitz 1972</a>.</span> </li> <li id="cite_note-FOOTNOTEHenneauxPerssonSpindel2008-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHenneauxPerssonSpindel2008_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHenneauxPerssonSpindel2008">Henneaux, Persson & Spindel 2008</a>.</span> </li> <li id="cite_note-FOOTNOTEHenneauxPerssonWesley2008-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHenneauxPerssonWesley2008_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHenneauxPerssonWesley2008">Henneaux, Persson & Wesley 2008</a>.</span> </li> <li id="cite_note-FOOTNOTEHenneaux2009-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHenneaux2009_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHenneaux2009">Henneaux 2009</a>.</span> </li> <li id="cite_note-FOOTNOTECornishLevin1997a-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECornishLevin1997a_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCornishLevin1997a">Cornish & Levin 1997a</a>.</span> </li> <li id="cite_note-FOOTNOTECornishLevin1997b-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECornishLevin1997b_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCornishLevin1997b">Cornish & Levin 1997b</a>.</span> </li> <li id="cite_note-FOOTNOTECornishLevin1997c-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECornishLevin1997c_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCornishLevin1997c">Cornish & Levin 1997c</a>.</span> </li> <li id="cite_note-LK-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-LK_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLifshitzKhalatnikov1963">Lifshitz & Khalatnikov 1963</a></span> </li> <li id="cite_note-LL-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-LL_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLandauLifshitz1988">Landau & Lifshitz 1988</a>, Ch. 97</span> </li> <li id="cite_note-FOOTNOTELandauLifshitz1988eq._(98.14)-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELandauLifshitz1988eq._(98.14)_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLandauLifshitz1988">Landau & Lifshitz 1988</a>, eq. (98.14).</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bianchi_classification&action=edit&section=9" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBelinskyKhalatnikovLifshitz1971" class="citation journal cs1"><a href="/wiki/Vladimir_A._Belinsky" class="mw-redirect" title="Vladimir A. Belinsky">Belinsky, Vladimir A.</a>; <a href="/wiki/Isaak_Markovich_Khalatnikov" class="mw-redirect" title="Isaak Markovich Khalatnikov">Khalatnikov, I.M.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E.M.</a> (1971). 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Bianchi, <i>Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti.</i> (On the spaces of three dimensions that admit a continuous group of movements.) Soc. Ital. Sci. Mem. di Mat. 11, 267 (1898) <a rel="nofollow" class="external text" href="http://ipsapp007.kluweronline.com/content/getfile/4728/60/13/abstract.htm">English translation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200218033450/http://ipsapp007.kluweronline.com/content/getfile/4728/60/13/abstract.htm">Archived</a> 2020-02-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornishLevin1997a" class="citation book cs1">Cornish, N.J.; Levin, J.J. (1997a). "The Mixmaster Universe is unambiguously chaotic". 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Harvey, Springer <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98564-6" title="Special:BookSources/0-387-98564-6">0-387-98564-6</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenneauxPerssonSpindel2008" class="citation journal cs1"><a href="/wiki/Marc_Henneaux" title="Marc Henneaux">Henneaux, Marc</a>; Persson, Daniel; Spindel, Philippe (2008). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255974">"Spacelike Singularities and Hidden Symmetries of Gravity"</a>. <i><a href="/wiki/Living_Reviews_in_Relativity" title="Living Reviews in Relativity">Living Reviews in Relativity</a></i>. <b>11</b> (1): 1. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0710.1818">0710.1818</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008LRR....11....1H">2008LRR....11....1H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.12942%2Flrr-2008-1">10.12942/lrr-2008-1</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255974">5255974</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28179821">28179821</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Reviews+in+Relativity&rft.atitle=Spacelike+Singularities+and+Hidden+Symmetries+of+Gravity&rft.volume=11&rft.issue=1&rft.pages=1&rft.date=2008&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5255974%23id-name%3DPMC&rft_id=info%3Abibcode%2F2008LRR....11....1H&rft_id=info%3Aarxiv%2F0710.1818&rft_id=info%3Apmid%2F28179821&rft_id=info%3Adoi%2F10.12942%2Flrr-2008-1&rft.aulast=Henneaux&rft.aufirst=Marc&rft.au=Persson%2C+Daniel&rft.au=Spindel%2C+Philippe&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5255974&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenneauxPerssonWesley2008" class="citation journal cs1"><a href="/wiki/Marc_Henneaux" title="Marc Henneaux">Henneaux, Marc</a>; Persson, Daniel; Wesley, Daniel (2008). "Coxeter group structure of cosmological billiards on compact spatial manifolds". <i>Journal of High Energy Physics</i>. <b>2008</b> (9): 052. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0805.3793">0805.3793</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JHEP...09..052H">2008JHEP...09..052H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1126-6708%2F2008%2F09%2F052">10.1088/1126-6708/2008/09/052</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1029-8479">1029-8479</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14135098">14135098</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Coxeter+group+structure+of+cosmological+billiards+on+compact+spatial+manifolds&rft.volume=2008&rft.issue=9&rft.pages=052&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14135098%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2008JHEP...09..052H&rft_id=info%3Aarxiv%2F0805.3793&rft.issn=1029-8479&rft_id=info%3Adoi%2F10.1088%2F1126-6708%2F2008%2F09%2F052&rft.aulast=Henneaux&rft.aufirst=Marc&rft.au=Persson%2C+Daniel&rft.au=Wesley%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenneaux2009" class="citation book cs1"><a href="/wiki/Marc_Henneaux" title="Marc Henneaux">Henneaux, Marc</a> (2009). "Kac-Moody algebras and the structure of cosmological singularities: A new light on the Belinskii-Khalatnikov-Lifshitz analysis". <i>Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity</i>. pp. 1–11. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0806.4670">0806.4670</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-87499-9_11">10.1007/978-0-387-87499-9_11</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-87498-2" title="Special:BookSources/978-0-387-87498-2"><bdi>978-0-387-87498-2</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18809715">18809715</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Kac-Moody+algebras+and+the+structure+of+cosmological+singularities%3A+A+new+light+on+the+Belinskii-Khalatnikov-Lifshitz+analysis&rft.btitle=Quantum+Mechanics+of+Fundamental+Systems%3A+The+Quest+for+Beauty+and+Simplicity&rft.pages=1-11&rft.date=2009&rft_id=info%3Aarxiv%2F0806.4670&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18809715%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-0-387-87499-9_11&rft.isbn=978-0-387-87498-2&rft.aulast=Henneaux&rft.aufirst=Marc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li>Robert T. Jantzen, <a rel="nofollow" class="external text" href="http://www34.homepage.villanova.edu/robert.jantzen/bianchi/">Bianchi classification of 3-geometries: original papers in translation</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJantzen2001" class="citation journal cs1">Jantzen, Robert T. (2001). "Spatially homogeneous dynamics: a unified picture". <i>Proc. Int. SCH. Phys. "E. Fermi" Course</i>. <b>LXXXVI</b>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0102035">gr-qc/0102035</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Int.+SCH.+Phys.+%22E.+Fermi%22+Course&rft.atitle=Spatially+homogeneous+dynamics%3A+a+unified+picture&rft.volume=LXXXVI&rft.date=2001&rft_id=info%3Aarxiv%2Fgr-qc%2F0102035&rft.aulast=Jantzen&rft.aufirst=Robert+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1988" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, Lev D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, Evgeny M.</a> (1988). <i>Classical Theory of Fields</i> (7th ed.). Moscow: <a href="/wiki/Nauka_(publisher)" title="Nauka (publisher)">Nauka</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-5-02-014420-0" title="Special:BookSources/978-5-02-014420-0"><bdi>978-5-02-014420-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Theory+of+Fields&rft.place=Moscow&rft.edition=7th&rft.pub=Nauka&rft.date=1988&rft.isbn=978-5-02-014420-0&rft.aulast=Landau&rft.aufirst=Lev+D.&rft.au=Lifshitz%2C+Evgeny+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span> Vol. 2 of the <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLifshitzKhalatnikov1963" class="citation journal cs1"><a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, Evgeny M.</a>; <a href="/wiki/Isaak_Markovich_Khalatnikov" class="mw-redirect" title="Isaak Markovich Khalatnikov">Khalatnikov, Isaak M.</a> (1963). <a rel="nofollow" class="external text" href="https://doi.org/10.3367%2FUFNr.0080.196307d.0391">"Проблемы релятивистской космологии"</a>. <i><a href="/wiki/Physics-Uspekhi" title="Physics-Uspekhi">Uspekhi Fizicheskikh Nauk</a></i>. <b>80</b> (7): 391–438. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3367%2FUFNr.0080.196307d.0391">10.3367/UFNr.0080.196307d.0391</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Uspekhi+Fizicheskikh+Nauk&rft.atitle=%D0%9F%D1%80%D0%BE%D0%B1%D0%BB%D0%B5%D0%BC%D1%8B+%D1%80%D0%B5%D0%BB%D1%8F%D1%82%D0%B8%D0%B2%D0%B8%D1%81%D1%82%D1%81%D0%BA%D0%BE%D0%B9+%D0%BA%D0%BE%D1%81%D0%BC%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D0%B8&rft.volume=80&rft.issue=7&rft.pages=391-438&rft.date=1963&rft_id=info%3Adoi%2F10.3367%2FUFNr.0080.196307d.0391&rft.aulast=Lifshitz&rft.aufirst=Evgeny+M.&rft.au=Khalatnikov%2C+Isaak+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.3367%252FUFNr.0080.196307d.0391&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span>; English translation in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLifshitz,_E.M.Khalatnikov1963" class="citation journal cs1">Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". <i><a href="/wiki/Advances_in_Physics" title="Advances in Physics">Advances in Physics</a></i>. <b>12</b> (46): 185. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963AdPhy..12..185L">1963AdPhy..12..185L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00018736300101283">10.1080/00018736300101283</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Physics&rft.atitle=Problems+in+the+Relativistic+Cosmology&rft.volume=12&rft.issue=46&rft.pages=185&rft.date=1963&rft_id=info%3Adoi%2F10.1080%2F00018736300101283&rft_id=info%3Abibcode%2F1963AdPhy..12..185L&rft.au=Lifshitz%2C+E.M.&rft.au=Khalatnikov%2C+I.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyanShepley1975" class="citation book cs1">Ryan, Michael P.; Shepley, Lawrence C. (1975). <a rel="nofollow" class="external text" href="https://inspirehep.net/files/0a60680b04f43d734a0ea07230a527a3"><i>Homogeneous Relativistic Cosmologies</i></a>. Princeton Series in Physics. Princeton, New Jersey: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691645209" title="Special:BookSources/9780691645209"><bdi>9780691645209</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homogeneous+Relativistic+Cosmologies&rft.place=Princeton%2C+New+Jersey&rft.series=Princeton+Series+in+Physics&rft.pub=Princeton+University+Press&rft.date=1975&rft.isbn=9780691645209&rft.aulast=Ryan&rft.aufirst=Michael+P.&rft.au=Shepley%2C+Lawrence+C.&rft_id=https%3A%2F%2Finspirehep.net%2Ffiles%2F0a60680b04f43d734a0ea07230a527a3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephaniKramerMacCallumHoenselaers2003" class="citation book cs1">Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003). <i>Exact Solutions of Einstein's Field Equations</i> (Second ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46136-8" title="Special:BookSources/978-0-521-46136-8"><bdi>978-0-521-46136-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exact+Solutions+of+Einstein%27s+Field+Equations&rft.edition=Second&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-46136-8&rft.aulast=Stephani&rft.aufirst=Hans&rft.au=Kramer%2C+Dietrich&rft.au=MacCallum%2C+Malcolm&rft.au=Hoenselaers%2C+Cornelius&rft.au=Herlt%2C+Eduard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWald1984" class="citation book cs1"><a href="/wiki/Robert_Wald" title="Robert Wald">Wald, Robert M.</a> (1984). <a href="/wiki/General_Relativity_(book)" title="General Relativity (book)"><i>General Relativity</i></a>. Chicago: University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-87033-2" title="Special:BookSources/0-226-87033-2"><bdi>0-226-87033-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity&rft.place=Chicago&rft.pub=University+of+Chicago+Press&rft.date=1984&rft.isbn=0-226-87033-2&rft.aulast=Wald&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABianchi+classification" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Bianchi_classification&oldid=1230343784">https://en.wikipedia.org/w/index.php?title=Bianchi_classification&oldid=1230343784</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Lie_algebras" title="Category:Lie algebras">Lie algebras</a></li><li><a href="/wiki/Category:Lie_groups" title="Category:Lie groups">Lie groups</a></li><li><a href="/wiki/Category:Physical_cosmology" title="Category:Physical cosmology">Physical cosmology</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 22 June 2024, at 05:30<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Bianchi classification - Wikipedia