Torsion tensor - Wikipedia

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class="vector-toc-list"> <li id="toc-Components_of_the_torsion_tensor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Components_of_the_torsion_tensor"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Components of the torsion tensor</span> </div> </a> <ul id="toc-Components_of_the_torsion_tensor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_torsion_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_torsion_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>The torsion form</span> </div> </a> <ul id="toc-The_torsion_form-sublist" class="vector-toc-list"> <li id="toc-Torsion_form_in_a_frame" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Torsion_form_in_a_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Torsion form in a frame</span> </div> </a> <ul id="toc-Torsion_form_in_a_frame-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Irreducible_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irreducible_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Irreducible decomposition</span> </div> </a> <ul id="toc-Irreducible_decomposition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Curvature_and_the_Bianchi_identities" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Curvature_and_the_Bianchi_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Curvature and the Bianchi identities</span> </div> </a> <button aria-controls="toc-Curvature_and_the_Bianchi_identities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_torsion_of_a_filament"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>The torsion of a filament</span> </div> </a> <ul id="toc-The_torsion_of_a_filament-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Torsion_and_vorticity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Torsion_and_vorticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Torsion and vorticity</span> </div> </a> <ul id="toc-Torsion_and_vorticity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geodesics_and_the_absorption_of_torsion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geodesics_and_the_absorption_of_torsion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geodesics and the absorption of torsion</span> </div> </a> <ul id="toc-Geodesics_and_the_absorption_of_torsion-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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul 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Available in 12 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-12" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">12 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Torsi%C3%B3_d%27una_connexi%C3%B3" title="Torsió d'una connexió – Catalan" lang="ca" hreflang="ca" data-title="Torsió d'una connexió" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Torsionstensor" title="Torsionstensor – German" lang="de" hreflang="de" data-title="Torsionstensor" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tensor_de_torsi%C3%B3n" title="Tensor de torsión – Spanish" lang="es" hreflang="es" data-title="Tensor de torsión" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D9%86%D8%B3%D9%88%D8%B1_%D8%AA%D8%A7%D8%A8%D8%AF%D8%A7%D8%B1" title="تانسور تابدار – Persian" lang="fa" hreflang="fa" data-title="تانسور تابدار" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tenseur_de_torsion" title="Tenseur de torsion – French" lang="fr" hreflang="fr" data-title="Tenseur de torsion" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B9%84%ED%8B%80%EB%A6%BC_%ED%85%90%EC%84%9C" title="비틀림 텐서 – Korean" lang="ko" hreflang="ko" data-title="비틀림 텐서" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Torsione_(geometria_differenziale)" title="Torsione (geometria differenziale) – Italian" lang="it" hreflang="it" data-title="Torsione (geometria differenziale)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8D%A9%E7%8E%87%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB" title="捩率テンソル – Japanese" lang="ja" hreflang="ja" data-title="捩率テンソル" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tensor_skr%C4%99cenia" title="Tensor skręcenia – Polish" lang="pl" hreflang="pl" data-title="Tensor skręcenia" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D1%87%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%B2%D1%8F%D0%B7%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Кручение связности – Russian" lang="ru" hreflang="ru" data-title="Кручение связности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80_%D0%BA%D1%80%D1%83%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Тензор кручення – Ukrainian" lang="uk" hreflang="uk" data-title="Тензор кручення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%89%AD%E7%8E%87%E5%BC%B5%E9%87%8F" title="扭率張量 – Chinese" lang="zh" hreflang="zh" data-title="扭率張量" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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searchaux" style="display:none">Manner of characterizing a twist or screw of a moving frame around a curve</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Torsion_(disambiguation)" class="mw-redirect mw-disambig" title="Torsion (disambiguation)">Torsion (disambiguation)</a> and <a href="/wiki/Torsion_field_(disambiguation)" class="mw-redirect mw-disambig" title="Torsion field (disambiguation)">Torsion field (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_development_with_torsion.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Circle_development_with_torsion.png/220px-Circle_development_with_torsion.png" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Circle_development_with_torsion.png/330px-Circle_development_with_torsion.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Circle_development_with_torsion.png/440px-Circle_development_with_torsion.png 2x" data-file-width="800" data-file-height="537" /></a><figcaption>Development of the unit circle in the Euclidean 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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>, with four different choices of flat connection preserving the Euclidean metric, defined 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 \nabla _{e_{i}}e_{j}=\tau \,e_{i}\times e_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{e_{i}}e_{j}=\tau \,e_{i}\times e_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799ea5dd4c28f1c9b1340812da3d3a3d181b9140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.957ex; height:2.843ex;" alt="{\displaystyle \nabla _{e_{i}}e_{j}=\tau \,e_{i}\times e_{j}}"></span>, 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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is a constant scalar, respectively: <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 \tau =0.01,0.1,0.5,1.0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>1.0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =0.01,0.1,0.5,1.0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8749dabf64128a5f1f337551f136ad95ff9d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.452ex; height:2.509ex;" alt="{\displaystyle \tau =0.01,0.1,0.5,1.0}"></span>. The resulting curves all have arc length <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span>, curvature <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, and respective torsion <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> (in the sense of <a href="/wiki/Frenet-Serret" class="mw-redirect" title="Frenet-Serret">Frenet-Serret</a>).</figcaption></figure> <p>In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the <b>torsion tensor</b> is a <a href="/wiki/Tensor" title="Tensor">tensor</a> that is associated to any <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a>. The torsion tensor is a <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear map</a> of two input 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 X,Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}"></span>, that produces an output 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 T(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc462da21ec8e84bc586f67adf0ce303c85ae1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.233ex; height:2.843ex;" alt="{\displaystyle T(X,Y)}"></span> representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are <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,Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}"></span>. It is <a href="/wiki/Skew_symmetric" class="mw-redirect" title="Skew symmetric">skew symmetric</a> in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how a <a href="/wiki/Screw" title="Screw">screw</a> moves in opposite ways when it is twisted in two directions. </p><p>Torsion is particularly useful in the study of the geometry of <a href="/wiki/Geodesic" title="Geodesic">geodesics</a>. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which <i>absorbs the torsion</i>, generalizing the <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a> to other, possibly non-metric situations (such as <a href="/wiki/Finsler_geometry" class="mw-redirect" title="Finsler geometry">Finsler geometry</a>). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the <a href="/wiki/Contorsion_tensor" title="Contorsion tensor">contorsion tensor</a>. Absorption of torsion also plays a fundamental role in the study of <a href="/wiki/G-structure" class="mw-redirect" title="G-structure">G-structures</a> and <a href="/wiki/Cartan%27s_equivalence_method" title="Cartan's equivalence method">Cartan's equivalence method</a>. Torsion is also useful in the study of unparametrized families of geodesics, via the associated <a href="/wiki/Projective_connection" title="Projective connection">projective connection</a>. In <a href="/wiki/Relativity_theory" class="mw-redirect" title="Relativity theory">relativity theory</a>, such ideas have been implemented in the form of <a href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan theory</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>M</i> be a manifold with an <a href="/wiki/Affine_connection" title="Affine connection">affine connection</a> on the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> (aka <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>) ∇. The <b>torsion tensor</b> (sometimes called the <i>Cartan</i> (<i>torsion</i>) <i>tensor</i>) of ∇ is the <a href="/wiki/Vector-valued_form" class="mw-redirect" title="Vector-valued form">vector-valued 2-form</a> defined on <a href="/wiki/Vector_field" title="Vector field">vector fields</a> <i>X</i> and <i>Y</i> by<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<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 T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>Y</mi> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>X</mi> <mo>−</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55760fe285c2f2c64f86ec729c788172c93bb0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.483ex; height:2.843ex;" alt="{\displaystyle T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]}"></span></dd></dl> <p>where <span class="nowrap">[<i>X</i>, <i>Y</i>]</span> is the <a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a> of two vector fields. By the <a href="/wiki/Leibniz_rule_(generalized_product_rule)" class="mw-redirect" title="Leibniz rule (generalized product rule)">Leibniz rule</a>, <i>T</i>(<i>fX</i>, <i>Y</i>) = <i>T</i>(<i>X</i>, <i>fY</i>) = <i>fT</i>(<i>X</i>, <i>Y</i>) for any <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a> <i>f</i>. So <i>T</i> is <a href="/wiki/Tensorial" class="mw-redirect" title="Tensorial">tensorial</a>, despite being defined in terms of the <a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">connection</a> which is a first order differential operator: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields. </p> <div class="mw-heading mw-heading3"><h3 id="Components_of_the_torsion_tensor">Components of the torsion tensor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=2" title="Edit section: Components of the torsion tensor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The components of the torsion 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 T^{c}{}_{ab}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{c}{}_{ab}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5163249363be8e4d177241255fc62642bcf126e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.471ex; height:2.676ex;" alt="{\displaystyle T^{c}{}_{ab}}"></span> in terms of a local <a href="/wiki/Basis_of_a_vector_space" class="mw-redirect" title="Basis of a vector space">basis</a> <span class="nowrap">(<b>e</b><sub>1</sub>, ..., <b>e</b><sub><i>n</i></sub>)</span> of <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> of the tangent bundle can be derived by setting <span class="nowrap"><i>X</i> = <b>e</b><sub><i>i</i></sub></span>, <span class="nowrap"><i>Y</i> = <b>e</b><sub><i>j</i></sub></span> and by introducing the commutator coefficients <span class="nowrap"><i>γ<sup>k</sup><sub>ij</sub></i><b>e</b><sub><i>k</i></sub> := [<b>e</b><sub><i>i</i></sub>, <b>e</b><sub><i>j</i></sub>]</span>. The components of the torsion are then<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<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 T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>:=</mo> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ca3fffa0eb44190f5e92e1654745fa398fcaa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.668ex; height:3.343ex;" alt="{\displaystyle T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}"></span></dd></dl> <p>Here <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 ^{k}}_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Gamma ^{k}}_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfbad24c9375f3eae294ed58691700b2161a08e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.019ex; height:3.343ex;" alt="{\displaystyle {\Gamma ^{k}}_{ij}}"></span> are the <a href="/wiki/Connection_coefficient" class="mw-redirect" title="Connection coefficient">connection coefficients</a> defining the connection. If the basis is <a href="/wiki/Holonomic_basis" title="Holonomic basis">holonomic</a> then the Lie brackets vanish, <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 ^{k}{}_{ij}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{k}{}_{ij}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141feeae97f555027c8319bcae789deda9eaf57c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.107ex; height:3.343ex;" alt="{\displaystyle \gamma ^{k}{}_{ij}=0}"></span>. So <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 T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>i</mi> <mi>j</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8247e0713e7e599e045a0a5c174d8dc6bd4e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.48ex; height:3.509ex;" alt="{\displaystyle T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}}"></span>. In particular (see below), while the <a href="/wiki/Geodesic" title="Geodesic">geodesic equations</a> determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part. </p> <div class="mw-heading mw-heading3"><h3 id="The_torsion_form">The torsion form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=3" title="Edit section: The torsion form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>torsion form</b>, an alternative characterization of torsion, applies to the <a href="/wiki/Frame_bundle" title="Frame bundle">frame bundle</a> F<i>M</i> of the manifold <i>M</i>. This <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> is equipped with a <a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">connection form</a> <i>ω</i>, a <b>gl</b>(<i>n</i>)-valued one-form which maps vertical vectors to the generators of the right action in <b>gl</b>(<i>n</i>) and equivariantly intertwines the right action of GL(<i>n</i>) on the tangent bundle of F<i>M</i> with the <a href="/wiki/Adjoint_representation_of_a_Lie_group" class="mw-redirect" title="Adjoint representation of a Lie group">adjoint representation</a> on <b>gl</b>(<i>n</i>). The frame bundle also carries a <a href="/wiki/Solder_form" title="Solder form">canonical one-form</a> θ, with values in <b>R</b><sup><i>n</i></sup>, defined at a frame <span class="nowrap"><i>u</i> ∈ F<sub>x</sub><i>M</i></span> (regarded as a linear function <span class="nowrap"><i>u</i> : <b>R</b><sup><i>n</i></sup> → T<sub>x</sub><i>M</i></span>) by<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<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 \theta (X)=u^{-1}(\pi _{*}(X))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (X)=u^{-1}(\pi _{*}(X))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/159194cb15cf1f4e63649797816054ac85e204e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.619ex; height:3.176ex;" alt="{\displaystyle \theta (X)=u^{-1}(\pi _{*}(X))}"></span></dd></dl> <p>where <span class="nowrap"><i>π</i>  : F<i>M</i> → <i>M</i></span> is the projection mapping for the principal bundle and <span class="nowrap"><i>π∗</i> </span> is its push-forward. The torsion form is then<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<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 \Theta =d\theta +\omega \wedge \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo>=</mo> <mi>d</mi> <mi>θ</mi> <mo>+</mo> <mi>ω</mi> <mo>∧</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta =d\theta +\omega \wedge \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f8f49bc09182a00c5d2d1b715659c395cf1bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.819ex; height:2.343ex;" alt="{\displaystyle \Theta =d\theta +\omega \wedge \theta .}"></span></dd></dl> <p>Equivalently, Θ = <i>Dθ</i>, where <i>D</i> is the <a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">exterior covariant derivative</a> determined by the connection. </p><p>The torsion form is a (horizontal) <a href="/wiki/Tensorial_form" class="mw-redirect" title="Tensorial form">tensorial form</a> with values in <b>R</b><sup><i>n</i></sup>, meaning that under the right action of <span class="nowrap"><i>g</i> ∈ GL(<i>n</i>)</span> it transforms <a href="/wiki/Equivariant" class="mw-redirect" title="Equivariant">equivariantly</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 R_{g}^{*}\Theta =g^{-1}\cdot \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi mathvariant="normal">Θ</mi> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{g}^{*}\Theta =g^{-1}\cdot \Theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a839604e4b6938bcc20a7cf653642250bf6393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.663ex; height:3.343ex;" alt="{\displaystyle R_{g}^{*}\Theta =g^{-1}\cdot \Theta }"></span></dd></dl> <p>where <i>g</i> acts on the right-hand side through its <a href="/wiki/Adjoint_representation" title="Adjoint representation">adjoint representation</a> on <b>R</b><sup><i>n</i></sup>. </p> <div class="mw-heading mw-heading4"><h4 id="Torsion_form_in_a_frame">Torsion form in a frame</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=4" title="Edit section: Torsion form in a frame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Connection_form" title="Connection form">connection form</a></div> <p>The torsion form may be expressed in terms of a <a href="/wiki/Connection_form" title="Connection form">connection form</a> on the base manifold <i>M</i>, written in a particular frame of the tangent bundle <span class="nowrap">(<b>e</b><sub>1</sub>, ..., <b>e</b><sub><i>n</i></sub>)</span>. The connection form expresses the exterior covariant derivative of these basic sections:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<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 D\mathbf {e} _{i}=\mathbf {e} _{j}{\omega ^{j}}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\mathbf {e} _{i}=\mathbf {e} _{j}{\omega ^{j}}_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e180989a1ec5082598344def9ab2c0abe1ce115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.985ex; height:3.343ex;" alt="{\displaystyle D\mathbf {e} _{i}=\mathbf {e} _{j}{\omega ^{j}}_{i}.}"></span></dd></dl> <p>The <a href="/wiki/Solder_form" title="Solder form">solder form</a> for the tangent bundle (relative to this frame) is the <a href="/wiki/Dual_basis" title="Dual basis">dual basis</a> <span class="nowrap"><i>θ<sup>i</sup></i> ∈ T<sup>∗</sup><i>M</i></span> of the <b>e</b><sub><i>i</i></sub>, so that <span class="nowrap"><i>θ<sup>i</sup></i>(<b>e</b><sub>j</sub>) = <i>δ<sup>i</sup><sub>j</sub></i></span> (the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>). Then the torsion 2-form has components </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 \Theta ^{k}=d\theta ^{k}+{\omega ^{k}}_{j}\wedge \theta ^{j}={T^{k}}_{ij}\theta ^{i}\wedge \theta ^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∧</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta ^{k}=d\theta ^{k}+{\omega ^{k}}_{j}\wedge \theta ^{j}={T^{k}}_{ij}\theta ^{i}\wedge \theta ^{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e51c30317158e2e4ec01adb8c4a0acfe45bde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.762ex; height:3.343ex;" alt="{\displaystyle \Theta ^{k}=d\theta ^{k}+{\omega ^{k}}_{j}\wedge \theta ^{j}={T^{k}}_{ij}\theta ^{i}\wedge \theta ^{j}.}"></span></dd></dl> <p>In the rightmost expression, </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 {T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left[\mathbf {e} _{i},\mathbf {e} _{j}\right]\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left[\mathbf {e} _{i},\mathbf {e} _{j}\right]\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003a8e60b5f698367219e5da83250e4a51b53c16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.816ex; height:3.509ex;" alt="{\displaystyle {T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left[\mathbf {e} _{i},\mathbf {e} _{j}\right]\right)}"></span></dd></dl> <p>are the frame-components of the torsion tensor, as given in the previous definition. </p><p>It can be easily shown that Θ<sup><i>i</i></sup> transforms tensorially in the sense that if a different frame </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 {\tilde {\mathbf {e} }}_{i}=\mathbf {e} _{j}{g^{j}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\mathbf {e} }}_{i}=\mathbf {e} _{j}{g^{j}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5886b2ab5c9eef15afef24f1783675d395de7f23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.086ex; height:3.343ex;" alt="{\displaystyle {\tilde {\mathbf {e} }}_{i}=\mathbf {e} _{j}{g^{j}}_{i}}"></span></dd></dl> <p>for some invertible matrix-valued function (<i>g</i><sup><i>j</i></sup><sub><i>i</i></sub>), then </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 {\tilde {\Theta }}^{i}={\left(g^{-1}\right)^{i}}_{j}\Theta ^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\Theta }}^{i}={\left(g^{-1}\right)^{i}}_{j}\Theta ^{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd420851f55a45b933dd6e38c62d0fcf46a0aad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:16.361ex; height:4.343ex;" alt="{\displaystyle {\tilde {\Theta }}^{i}={\left(g^{-1}\right)^{i}}_{j}\Theta ^{j}.}"></span></dd></dl> <p>In other terms, Θ is a tensor of type <span class="nowrap">(1, 2)</span> (carrying one contravariant and two covariant indices). </p><p>Alternatively, the solder form can be characterized in a frame-independent fashion as the T<i>M</i>-valued one-form <i>θ</i> on <i>M</i> corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism <span class="nowrap">End(T<i>M</i>) ≈ T<i>M</i> ⊗ T<sup>∗</sup><i>M</i></span>. Then the torsion 2-form is a section </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 \Theta \in {\text{Hom}}\left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hom</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mi>M</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta \in {\text{Hom}}\left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/217517cb08969d907bb163a1390b90c3bff06b8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.917ex; height:4.843ex;" alt="{\displaystyle \Theta \in {\text{Hom}}\left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right)}"></span></dd></dl> <p>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 \Theta =D\theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo>=</mo> <mi>D</mi> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta =D\theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a227f32ecdded669c5005c87b9fbc4446cf967aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.568ex; height:2.509ex;" alt="{\displaystyle \Theta =D\theta ,}"></span></dd></dl> <p>where <i>D</i> is the <a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">exterior covariant derivative</a>. (See <a href="/wiki/Connection_form" title="Connection form">connection form</a> for further details.) </p> <div class="mw-heading mw-heading3"><h3 id="Irreducible_decomposition">Irreducible decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=5" title="Edit section: Irreducible decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The torsion tensor can be decomposed into two <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible</a> parts: a <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace-free</a> part and another part which contains the trace terms. Using the <a href="/wiki/Index_notation" title="Index notation">index notation</a>, the trace of <i>T</i> 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 a_{i}=T^{k}{}_{ik},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=T^{k}{}_{ik},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f952291614d5dc78cec878add17b44dbb404861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.24ex; height:3.009ex;" alt="{\displaystyle a_{i}=T^{k}{}_{ik},}"></span></dd></dl> <p>and the trace-free part 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 B^{i}{}_{jk}=T^{i}{}_{jk}+{\frac {1}{n-1}}\delta ^{i}{}_{j}a_{k}-{\frac {1}{n-1}}\delta ^{i}{}_{k}a_{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{i}{}_{jk}=T^{i}{}_{jk}+{\frac {1}{n-1}}\delta ^{i}{}_{j}a_{k}-{\frac {1}{n-1}}\delta ^{i}{}_{k}a_{j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78975d674979433d3a83aa10041fc5505c2f6a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.672ex; height:5.343ex;" alt="{\displaystyle B^{i}{}_{jk}=T^{i}{}_{jk}+{\frac {1}{n-1}}\delta ^{i}{}_{j}a_{k}-{\frac {1}{n-1}}\delta ^{i}{}_{k}a_{j},}"></span></dd></dl> <p>where <i>δ<sup>i</sup><sub>j</sub></i> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. </p><p>Intrinsically, one has </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 T\in \operatorname {Hom} \left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>∈</mo> <mi>Hom</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mi>M</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in \operatorname {Hom} \left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21366119fd4cf75eafc42d49463eb3c89fc41bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.005ex; height:4.843ex;" alt="{\displaystyle T\in \operatorname {Hom} \left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).}"></span></dd></dl> <p>The trace of <i>T</i>, tr <i>T</i>, is an element of T<sup>∗</sup><i>M</i> defined as follows. For each vector fixed <span class="nowrap"><i>X</i> ∈ T<i>M</i></span>, <i>T</i> defines an element <i>T</i>(<i>X</i>) of <span class="nowrap">Hom(T<i>M</i>, T<i>M</i>)</span> via </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 T(X):Y\mapsto T(X\wedge Y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">↦</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∧</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X):Y\mapsto T(X\wedge Y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e133e071d3b20a562baf82db1eaa4c85c264959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.178ex; height:2.843ex;" alt="{\displaystyle T(X):Y\mapsto T(X\wedge Y).}"></span></dd></dl> <p>Then (tr <i>T</i>)(<i>X</i>) is defined as the trace of this endomorphism. That 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 (\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>tr</mi> <mspace width="thinmathspace" /> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa81ab53f36412cf0f6347e2d6a106ce3686287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.058ex; height:3.843ex;" alt="{\displaystyle (\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).}"></span></dd></dl> <p>The trace-free part of <i>T</i> is then </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 T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>ι</mi> <mo stretchy="false">(</mo> <mi>tr</mi> <mspace width="thinmathspace" /> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c5d8f042226f1a637bb09bb2956312baf76e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.727ex; height:5.343ex;" alt="{\displaystyle T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),}"></span></dd></dl> <p>where <i>ι</i> denotes the <a href="/wiki/Interior_product" title="Interior product">interior product</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Curvature_and_the_Bianchi_identities">Curvature and the Bianchi identities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=6" title="Edit section: Curvature and the Bianchi identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">curvature tensor</a> of ∇ is a mapping <span class="nowrap">T<i>M</i> × T<i>M</i> → End(T<i>M</i>)</span> defined on vector fields <i>X</i>, <i>Y</i>, and <i>Z</i> 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(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mi>Z</mi> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mi>Z</mi> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>Z</mi> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mi>Z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4c06786c30c0e5937bbce502e4f812c73026b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.683ex; height:3.176ex;" alt="{\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.}"></span></dd></dl> <p>For vectors at a point, this definition is independent of how the vectors are extended to vector fields away from the point (thus it defines a tensor, much like the torsion). </p><p>The <b>Bianchi identities</b> relate the curvature and torsion as follows.<sup id="cite_ref-FOOTNOTEKobayashiNomizu1963Volume_1,_Proposition_III.5.2_6-0" class="reference"><a href="#cite_note-FOOTNOTEKobayashiNomizu1963Volume_1,_Proposition_III.5.2-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Let <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 {\mathfrak {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91d521f12ee3c00ee2fc7ab16af9ea17d915a750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {S}}}"></span> denote the <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cyclic sum</a> over <i>X</i>, <i>Y</i>, and <i>Z</i>. For instance, </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 {\mathfrak {S}}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mi>Z</mi> </mrow> <mo>)</mo> </mrow> <mo>:=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mi>Z</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mi>X</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca979419f7e55146b2e1979de7ebec8bdc8f3614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.134ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {S}}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.}"></span></dd></dl> <p>Then the following identities hold </p> <ol><li><b>Bianchi's first identity:</b> <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 {\mathfrak {S}}\left(R\left(X,Y\right)Z\right)={\mathfrak {S}}\left(T\left(T(X,Y),Z\right)+\left(\nabla _{X}T\right)\left(Y,Z\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mi>Z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>Z</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>T</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}\left(R\left(X,Y\right)Z\right)={\mathfrak {S}}\left(T\left(T(X,Y),Z\right)+\left(\nabla _{X}T\right)\left(Y,Z\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd49e7ff192a7c5433fecc4dd9f9158c1d77a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.478ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {S}}\left(R\left(X,Y\right)Z\right)={\mathfrak {S}}\left(T\left(T(X,Y),Z\right)+\left(\nabla _{X}T\right)\left(Y,Z\right)\right)}"></span></dd></dl></li> <li><b>Bianchi's second identity:</b> <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 {\mathfrak {S}}\left(\left(\nabla _{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>Z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}\left(\left(\nabla _{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/624a5c12e2b5780b3de894246d9484345ceceff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.732ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {S}}\left(\left(\nabla _{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0}"></span></dd></dl></li></ol> <div class="mw-heading mw-heading3"><h3 id="The_curvature_form_and_Bianchi_identities">The curvature form and Bianchi identities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=7" title="Edit section: The curvature form and Bianchi identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Curvature_form" title="Curvature form">curvature form</a> is the <b>gl</b>(<i>n</i>)-valued 2-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 \Omega =D\omega =d\omega +\omega \wedge \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi>D</mi> <mi>ω</mi> <mo>=</mo> <mi>d</mi> <mi>ω</mi> <mo>+</mo> <mi>ω</mi> <mo>∧</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =D\omega =d\omega +\omega \wedge \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac32973a2d6bff3e90ddbc4d2c939a771520548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.221ex; height:2.343ex;" alt="{\displaystyle \Omega =D\omega =d\omega +\omega \wedge \omega }"></span></dd></dl> <p>where, again, <i>D</i> denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi identities are<sup id="cite_ref-FOOTNOTEKobayashiNomizu1963Volume_1,_III.2_7-0" class="reference"><a href="#cite_note-FOOTNOTEKobayashiNomizu1963Volume_1,_III.2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ol><li><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 D\Theta =\Omega \wedge \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi mathvariant="normal">Θ</mi> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo>∧</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\Theta =\Omega \wedge \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d0743944b1036b86956cfed7719855ac414b4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.182ex; height:2.176ex;" alt="{\displaystyle D\Theta =\Omega \wedge \theta }"></span></li> <li><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 D\Omega =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\Omega =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7532fb8c32b0b241e154c06b2ea1572af542c6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.51ex; height:2.176ex;" alt="{\displaystyle D\Omega =0.}"></span></li></ol> <p>Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a point <i>u</i> of F<sub>x</sub><i>M</i>, one has<sup id="cite_ref-FOOTNOTEKobayashiNomizu1963Volume_1,_III.5_8-0" class="reference"><a href="#cite_note-FOOTNOTEKobayashiNomizu1963Volume_1,_III.5-8"><span class="cite-bracket">[</span>8<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 {\begin{aligned}R(X,Y)Z&=u\left(2\Omega \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right)\left(u^{-1}(Z)\right),\\T(X,Y)&=u\left(2\Theta \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>R</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mi>Z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi mathvariant="normal">Θ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R(X,Y)Z&=u\left(2\Omega \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right)\left(u^{-1}(Z)\right),\\T(X,Y)&=u\left(2\Theta \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d47b695bf0c8b93c90ffd6f921f3e9868dfa4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:49.538ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}R(X,Y)Z&=u\left(2\Omega \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right)\left(u^{-1}(Z)\right),\\T(X,Y)&=u\left(2\Theta \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right),\end{aligned}}}"></span></dd></dl> <p>where again <span class="nowrap"><i>u</i> : <b>R</b><sup><i>n</i></sup> → T<sub>x</sub><i>M</i></span> is the function specifying the frame in the fibre, and the choice of lift of the vectors via π<sup>−1</sup> is irrelevant since the curvature and torsion forms are horizontal (they vanish on the ambiguous vertical vectors). </p> <div class="mw-heading mw-heading2"><h2 id="Characterizations_and_interpretations">Characterizations and interpretations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=8" title="Edit section: Characterizations and interpretations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The torsion is a manner of characterizing the amount of slipping or twisting that a plane does when rolling along a surface or higher dimensional <a href="/wiki/Affine_manifold" title="Affine manifold">affine manifold</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>For example, consider rolling a plane along a small circle drawn on a sphere. If the plane does not slip or twist, then when the plane is rolled all the way along the circle, it will also trace a circle in the plane. It turns out that the plane will have rotated (despite there being no twist whilst rolling it), an effect due to the <a href="/wiki/Curvature" title="Curvature">curvature</a> of the sphere. But the curve traced out will still be a circle, and so in particular a closed curve that begins and ends at the same point. On the other hand, if the plane were rolled along the sphere, but it was allowed it to slip or twist in the process, then the path the circle traces on the plane could be a much more general curve that need not even be closed. The torsion is a way to quantify this additional slipping and twisting while rolling a plane along a curve. </p><p>Thus the torsion tensor can be intuitively understood by taking a small parallelogram circuit with sides given by vectors <i>v</i> and <i>w</i>, in a space and rolling the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> along each of the four sides of the parallelogram, marking the point of contact as it goes. When the circuit is completed, the marked curve will have been displaced out of the plane of the parallelogram by a vector, denoted <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 T(v,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v,w)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6978758887b0d8e5a6bd76901954cd8b12b87a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.271ex; height:2.843ex;" alt="{\displaystyle T(v,w)}"></span>. Thus the torsion tensor is a tensor: a (bilinear) function of two input vectors <i>v</i> and <i>w</i> that produces an output 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 T(v,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v,w)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6978758887b0d8e5a6bd76901954cd8b12b87a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.271ex; height:2.843ex;" alt="{\displaystyle T(v,w)}"></span>. It is <a href="/wiki/Skew_symmetric" class="mw-redirect" title="Skew symmetric">skew symmetric</a> in the arguments <i>v</i> and <i>w</i>, a reflection of the fact that traversing the circuit in the opposite sense undoes the original displacement, in much the same way that twisting a <a href="/wiki/Screw" title="Screw">screw</a> in opposite directions displaces the screw in opposite ways. The torsion tensor thus is related to, although distinct from, the <a href="/wiki/Torsion_of_curves" class="mw-redirect" title="Torsion of curves">torsion of a curve</a>, as it appears in the <a href="/wiki/Frenet%E2%80%93Serret_formulas" title="Frenet–Serret formulas">Frenet–Serret formulas</a>: the torsion of a connection measures a dislocation of a developed curve out of its plane, while the torsion of a curve is also a dislocation out of its <a href="/wiki/Osculating_plane" title="Osculating plane">osculating plane</a>. In the geometry of surfaces, the <i>geodesic torsion</i> describes how a surface twists about a curve on the surface. The companion notion of <a href="/wiki/Curvature" title="Curvature">curvature</a> measures how moving frames roll along a curve without slipping or twisting. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=9" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the (flat) <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</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 M=\mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f43abaf9eea56410f58a7ab6cbeb882b82bc376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.273ex; height:2.676ex;" alt="{\displaystyle M=\mathbb {R} ^{3}}"></span>. On it, we put a connection that is flat, but with non-zero torsion, defined on the standard Euclidean frame <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_{1},e_{2},e_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},e_{2},e_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35821963b9761d80f5b4079f1ba179ac101da80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.481ex; height:2.009ex;" alt="{\displaystyle e_{1},e_{2},e_{3}}"></span> by the (Euclidean) <a href="/wiki/Cross_product" title="Cross product">cross product</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{e_{i}}e_{j}=e_{i}\times e_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{e_{i}}e_{j}=e_{i}\times e_{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ef755fa23bddcd2d67917e2dc6a565b8cfe403" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.014ex; height:2.843ex;" alt="{\displaystyle \nabla _{e_{i}}e_{j}=e_{i}\times e_{j}.}"></span> Consider now the parallel transport of 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 e_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4045b5c7cee9bd0681153bbb077489b13269355e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}"></span> along 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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e81caf3d4bcb929315801cbabc83543829484ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}"></span> axis, starting at the origin. The parallel vector field <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(x)=a(x)e_{2}+b(x)e_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(x)=a(x)e_{2}+b(x)e_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b7c9200b2bac816adfe0926f241b7100a0b015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.839ex; height:2.843ex;" alt="{\displaystyle X(x)=a(x)e_{2}+b(x)e_{3}}"></span> thus satisfies <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(0)=e_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(0)=e_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cada6dd81cc71b78bd10b184fdd152fa786842ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.188ex; height:2.843ex;" alt="{\displaystyle X(0)=e_{2}}"></span>, and the differential equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}0={\dot {X}}&=\nabla _{e_{1}}X={\dot {a}}e_{2}+{\dot {b}}e_{3}+ae_{1}\times e_{2}+be_{1}\times e_{3}\\&=({\dot {a}}-b)e_{2}+({\dot {b}}+a)e_{3}.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}0={\dot {X}}&=\nabla _{e_{1}}X={\dot {a}}e_{2}+{\dot {b}}e_{3}+ae_{1}\times e_{2}+be_{1}\times e_{3}\\&=({\dot {a}}-b)e_{2}+({\dot {b}}+a)e_{3}.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23400a100a23e2063de6969c488c1c7735120f2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:52.261ex; height:6.843ex;" alt="{\displaystyle {\begin{array}{rl}0={\dot {X}}&=\nabla _{e_{1}}X={\dot {a}}e_{2}+{\dot {b}}e_{3}+ae_{1}\times e_{2}+be_{1}\times e_{3}\\&=({\dot {a}}-b)e_{2}+({\dot {b}}+a)e_{3}.\end{array}}}"></span> Thus <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 {\dot {a}}=b,{\dot {b}}=-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>b</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {a}}=b,{\dot {b}}=-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0054338ec9d2a67951524066e1373094f16bae4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.659ex; height:3.009ex;" alt="{\displaystyle {\dot {a}}=b,{\dot {b}}=-a}"></span>, and the solution is <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=\cos x\,e_{2}-\sin x\,e_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\cos x\,e_{2}-\sin x\,e_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e86fb3ab8c122bffe64f25fd859b732aa726d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.369ex; height:2.509ex;" alt="{\displaystyle X=\cos x\,e_{2}-\sin x\,e_{3}}"></span>. </p><p>Now the tip of 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, as it is transported along 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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e81caf3d4bcb929315801cbabc83543829484ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}"></span> axis traces out the helix <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,e_{1}+\cos x\,e_{2}-\sin x\,e_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,e_{1}+\cos x\,e_{2}-\sin x\,e_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c40335f8c430e4bc21c5b611a64e622593ff57e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.632ex; height:2.509ex;" alt="{\displaystyle x\,e_{1}+\cos x\,e_{2}-\sin x\,e_{3}.}"></span> Thus we see that, in the presence of torsion, parallel transport tends to twist a frame around the direction of motion, analogously to the role played by torsion in the classical <a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">differential geometry of curves</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Development">Development</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=10" title="Edit section: Development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One interpretation of the torsion involves the development of a curve.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Suppose that a piecewise smooth closed loop <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 :[0,1]\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma :[0,1]\to M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0066953642fb00abb394327531cea098815cd1c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.908ex; height:2.843ex;" alt="{\displaystyle \gamma :[0,1]\to M}"></span> is given, based at the point <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\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}"></span>, 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 \gamma (0)=\gamma (1)=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (0)=\gamma (1)=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de848718109d270e79dfdf7d286e67b82f66c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.835ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=\gamma (1)=p}"></span>. We assume that <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is homotopic to zero. The curve can be developed into the tangent space at <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> in the following manner. Let <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 \theta ^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842a8b679590096930aa1d7f75a6cea3a31340b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.89ex; height:2.676ex;" alt="{\displaystyle \theta ^{i}}"></span> be a parallel coframe 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, and let <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> be the coordinates on <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 T_{p}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b20e0304a65ead2cafb33412a383b34fe527ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}"></span> induced 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 \theta ^{i}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{i}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09acf148bb9d1ac598ea610a8816d7f77401499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.869ex; height:3.176ex;" alt="{\displaystyle \theta ^{i}(p)}"></span>. A development 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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is a curve <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 {\tilde {\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b090b5a4f599c62fc511fa45ba936b140c7541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.676ex;" alt="{\displaystyle {\tilde {\gamma }}}"></span> in <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 T_{p}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b20e0304a65ead2cafb33412a383b34fe527ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}"></span> whose coordinates <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}=x^{i}(t)}"> <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> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}=x^{i}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4a30c0955317022c5198c09a0028f7a92388e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.006ex; height:3.176ex;" alt="{\displaystyle x^{i}=x^{i}(t)}"></span> sastify the differential equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{i}=\gamma ^{*}\theta ^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx^{i}=\gamma ^{*}\theta ^{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25cf65015436ee1855b331c2c26f037b32af3503" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.315ex; height:3.176ex;" alt="{\displaystyle dx^{i}=\gamma ^{*}\theta ^{i}.}"></span> If the torsion is zero, then the developed curve <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 {\tilde {\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b090b5a4f599c62fc511fa45ba936b140c7541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.676ex;" alt="{\displaystyle {\tilde {\gamma }}}"></span> is also a closed loop (so that <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 {\tilde {\gamma }}(0)={\tilde {\gamma }}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}(0)={\tilde {\gamma }}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/703c85952c7f4115475de3156d4c7366cb4cd604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.567ex; height:2.843ex;" alt="{\displaystyle {\tilde {\gamma }}(0)={\tilde {\gamma }}(1)}"></span>). On the other hand, if the torsion is non-zero, then the developed curve may not be closed, so that <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 {\tilde {\gamma }}(0)\not ={\tilde {\gamma }}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}(0)\not ={\tilde {\gamma }}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/472324e29c23e5cc7d7ba7a0f981f6ab1430ccd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.567ex; height:2.843ex;" alt="{\displaystyle {\tilde {\gamma }}(0)\not ={\tilde {\gamma }}(1)}"></span>. Thus the development of a loop in the presence of torsion can become dislocated, analogously to a <a href="/wiki/Screw_dislocation" class="mw-redirect" title="Screw dislocation">screw dislocation</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The foregoing considerations can be made more quantitative by considering a small parallelogram, originating at the point <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\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad2c18a15749505c928763cd4fdb56f4982816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}"></span>, with sides <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,w\in T_{p}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>∈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,w\in T_{p}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd99b11e1c18f437b4fa006ce407e8303c8a035d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.525ex; height:2.843ex;" alt="{\displaystyle v,w\in T_{p}M}"></span>. Then the tangent bivector to the parallelogram is <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\wedge w\in \Lambda ^{2}T_{p}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∧</mo> <mi>w</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\wedge w\in \Lambda ^{2}T_{p}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d4929db0658fc3b9dc6904e6eacd40026e494d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.741ex; height:3.343ex;" alt="{\displaystyle v\wedge w\in \Lambda ^{2}T_{p}M}"></span>. The development of this parallelogram, using the connection, is no longer closed in general, and the displacement in going around the loop is translation by 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 \Theta (v,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (v,w)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52db2a23823f716ad054d6b55a858912383814b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.443ex; height:2.843ex;" alt="{\displaystyle \Theta (v,w)}"></span>, 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 \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }"></span> is the torsion tensor, up to higher order terms in <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,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.009ex;" alt="{\displaystyle v,w}"></span>. This displacement is directly analogous to the <a href="/wiki/Burgers_vector" title="Burgers vector">Burgers vector</a> of crystallography.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>More generally, one can also transport a <a href="/wiki/Moving_frame" title="Moving frame">moving frame</a> along the curve <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 {\tilde {\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b090b5a4f599c62fc511fa45ba936b140c7541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.676ex;" alt="{\displaystyle {\tilde {\gamma }}}"></span>. The <i>linear</i> transformation that the frame undergoes between <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 t=0,t=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0,t=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c029117c742b4951e572f6991e0f5cd33a27abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.235ex; height:2.509ex;" alt="{\displaystyle t=0,t=1}"></span> is then determined by the curvature of the connection. Together, the linear transformation of the frame and the translation of the starting point from <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 {\tilde {\gamma }}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4b8822e3074965064164da7a8ed892b70c7861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.234ex; height:2.843ex;" alt="{\displaystyle {\tilde {\gamma }}(0)}"></span> to <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 {\tilde {\gamma }}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\gamma }}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7072a440ed521ab98866412fbfc28402913a194f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.234ex; height:2.843ex;" alt="{\displaystyle {\tilde {\gamma }}(1)}"></span> comprise the <a href="/wiki/Holonomy" title="Holonomy">holonomy</a> of the connection. </p> <div class="mw-heading mw-heading3"><h3 id="The_torsion_of_a_filament">The torsion of a filament</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=11" title="Edit section: The torsion of a filament"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Materials_science" title="Materials science">materials science</a>, and especially <a href="/wiki/Elasticity_theory" class="mw-redirect" title="Elasticity theory">elasticity theory</a>, ideas of torsion also play an important role. One problem models the growth of vines, focusing on the question of how vines manage to twist around objects.<sup id="cite_ref-FOOTNOTEGorielyRobertson-TessiTaborVandiver2006_14-0" class="reference"><a href="#cite_note-FOOTNOTEGorielyRobertson-TessiTaborVandiver2006-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The vine itself is modeled as a pair of elastic filaments twisted around one another. In its energy-minimizing state, the vine naturally grows in the shape of a <a href="/wiki/Helix" title="Helix">helix</a>. But the vine may also be stretched out to maximize its extent (or length). In this case, the torsion of the vine is related to the torsion of the pair of filaments (or equivalently the surface torsion of the ribbon connecting the filaments), and it reflects the difference between the length-maximizing (geodesic) configuration of the vine and its energy-minimizing configuration. </p> <div class="mw-heading mw-heading3"><h3 id="Torsion_and_vorticity">Torsion and vorticity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=12" title="Edit section: Torsion and vorticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, torsion is naturally associated to <a href="/wiki/Vortex_line" class="mw-redirect" title="Vortex line">vortex lines</a>. </p><p>Suppose that a connection <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is given in three dimensions, with curvature 2-form <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 \Omega _{a}^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{a}^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89aa1b05f7701871e1cdcc0aae49af6595036876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.78ex; height:2.843ex;" alt="{\displaystyle \Omega _{a}^{b}}"></span> and torsion 2-form <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 \Theta ^{a}=D\theta ^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mi>D</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta ^{a}=D\theta ^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef62823d54b01387a224049ae6b90eb55f02fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.125ex; height:2.343ex;" alt="{\displaystyle \Theta ^{a}=D\theta ^{a}}"></span>. Let <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 _{abc}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{abc}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a69f2df65c9d952a6404bccb971e06f03a3f58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.675ex; height:2.176ex;" alt="{\displaystyle \eta _{abc}}"></span> be the skew-symmetric <a href="/wiki/Levi-Civita_tensor" class="mw-redirect" title="Levi-Civita tensor">Levi-Civita tensor</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{a}={\tfrac {1}{2}}\eta _{abc}\wedge \Omega ^{bc},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>∧</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{a}={\tfrac {1}{2}}\eta _{abc}\wedge \Omega ^{bc},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5145ad40c33b05b9c157e73909167c30d49cab1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.93ex; height:3.509ex;" alt="{\displaystyle t_{a}={\tfrac {1}{2}}\eta _{abc}\wedge \Omega ^{bc},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{ab}=-\eta _{abc}\wedge \Theta ^{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msub> <mo>∧</mo> <msup> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{ab}=-\eta _{abc}\wedge \Theta ^{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6bad3b9251d7fe7fdaceaac60bc029be15525b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.461ex; height:2.843ex;" alt="{\displaystyle s_{ab}=-\eta _{abc}\wedge \Theta ^{c}.}"></span> Then the Bianchi identities The Bianchi identities are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\Omega _{b}^{a}=0,\quad D\Theta ^{a}=\Omega _{b}^{a}\wedge \theta ^{b}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>D</mi> <msup> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>∧</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\Omega _{b}^{a}=0,\quad D\Theta ^{a}=\Omega _{b}^{a}\wedge \theta ^{b}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d755753b6aa2184f2400596aede8ce0ea54f6e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.292ex; height:3.343ex;" alt="{\displaystyle D\Omega _{b}^{a}=0,\quad D\Theta ^{a}=\Omega _{b}^{a}\wedge \theta ^{b}.}"></span> imply that <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 Dt_{a}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Dt_{a}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e134ae027cd11c37888eb5e6d7f0d82fa3a56d2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.127ex; height:2.509ex;" alt="{\displaystyle Dt_{a}=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ds_{ab}=\theta _{a}\wedge t_{b}-\theta _{b}\wedge t_{a}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>∧</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>∧</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ds_{ab}=\theta _{a}\wedge t_{b}-\theta _{b}\wedge t_{a}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be59c064b9e54950eb88ccd9a71371f4a6e019db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.512ex; height:2.509ex;" alt="{\displaystyle Ds_{ab}=\theta _{a}\wedge t_{b}-\theta _{b}\wedge t_{a}.}"></span> These are the equations satisfied by an equilibrium continuous medium with moment 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 s_{ab}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{ab}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03f0696ab9462e37b39efc6595e122531d08cf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.898ex; height:2.009ex;" alt="{\displaystyle s_{ab}}"></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Geodesics_and_the_absorption_of_torsion">Geodesics and the absorption of torsion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_tensor&action=edit&section=13" title="Edit section: Geodesics and the absorption of torsion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <i>γ</i>(<i>t</i>) is a curve on <i>M</i>. Then <i>γ</i> is an <b>affinely parametrized geodesic</b> provided 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 \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85345c0b0be91d3df5147dfccffbc98a3a7b99ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.106ex; height:3.343ex;" alt="{\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}"></span></dd></dl> <p>for all time <i>t</i> in the domain of <i>γ</i>. (Here the dot denotes differentiation with respect to <i>t</i>, which associates with γ the tangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at time <span class="nowrap"><i>t</i> = 0</span>, <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 {\dot {\gamma }}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\gamma }}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1479c5439158b0681da11338d0ef23f5506448f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.234ex; height:2.843ex;" alt="{\displaystyle {\dot {\gamma }}(0)}"></span>. </p><p>One application of the torsion of a connection involves the <a href="/wiki/Geodesic_spray" class="mw-redirect" title="Geodesic spray">geodesic spray</a> of the connection: roughly the family of all affinely parametrized geodesics. Torsion is the ambiguity of classifying connections in terms of their geodesic sprays: </p> <ul><li>Two connections ∇ and ∇′ which have the same affinely parametrized geodesics (i.e., the same geodesic spray) differ only by torsion.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li></ul> <p>More precisely, if <i>X</i> and <i>Y</i> are a pair of tangent vectors at <span class="nowrap"><i>p</i> ∈ <i>M</i></span>, then let </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 \Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>−</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7b4b56b34cb1415409190cc905e171a20319d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.154ex; height:3.343ex;" alt="{\displaystyle \Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}}"></span></dd></dl> <p>be the difference of the two connections, calculated in terms of arbitrary extensions of <i>X</i> and <i>Y</i> away from <i>p</i>. By the <a href="/wiki/Product_rule" title="Product rule">Leibniz product rule</a>, one sees that Δ does not actually depend on how <i>X</i> and <i>Y</i><span class="nowrap" style="padding-left:0.15em;">′</span> are extended (so it defines a tensor on <i>M</i>). Let <i>S</i> and <i>A</i> be the symmetric and alternating parts of Δ: </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 S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b957537fd248f638a792d1baf5a1f5a2cb5f40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.954ex; height:3.509ex;" alt="{\displaystyle S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)}"></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(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6704108ef52a12d8c3a4464048f8896d4164002c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.198ex; height:3.509ex;" alt="{\displaystyle A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)}"></span></dd></dl> <p>Then </p> <ul><li><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(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb41ba7267c7eb87da5c63b0a1ec75a60c165c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.367ex; height:3.509ex;" alt="{\displaystyle A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)}"></span> is the difference of the torsion tensors.</li> <li>∇ and ∇′ define the same families of affinely parametrized geodesics if and only if <span class="nowrap"><i>S</i>(<i>X</i>, <i>Y</i>) = 0</span>.</li></ul> <p>In other words, the symmetric part of the difference of two connections determines whether they have the same parametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two connections. Another consequence is: </p> <ul><li>Given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. The difference between these two connections is in fact a tensor, the <a href="/wiki/Contorsion_tensor" title="Contorsion tensor">contorsion tensor</a>.</li></ul> <p>This is a generalization of the <a href="/wiki/Fundamental_theorem_of_Riemannian_geometry" title="Fundamental theorem of Riemannian geometry">fundamental theorem of Riemannian geometry</a> to general affine (possibly non-metric) connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known as <b>absorption of torsion</b>, and it is one of the stages of <a href="/wiki/Cartan%27s_equivalence_method" title="Cartan's equivalence method">Cartan's equivalence method</a>. </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=Torsion_tensor&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Contorsion_tensor" title="Contorsion tensor">Contorsion tensor</a></li> <li><a href="/wiki/Curtright_field" title="Curtright field">Curtright field</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Curvature tensor</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Torsion_coefficient_(topology)" class="mw-redirect" title="Torsion coefficient (topology)">Torsion coefficient</a></li> <li><a href="/wiki/Torsion_of_curves" class="mw-redirect" title="Torsion of curves">Torsion of curves</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=Torsion_tensor&action=edit&section=15" 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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu (1963)</a>, Chapter III, Theorem 5.1</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu (1963)</a>, Chapter III, Proposition 7.6</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu (1963)</a>, Chapter III, Section 2</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu (1963)</a>, Chapter III, Theorem 2.4</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu (1963)</a>, Chapter III, Section 7</span> </li> <li id="cite_note-FOOTNOTEKobayashiNomizu1963Volume_1,_Proposition_III.5.2-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKobayashiNomizu1963Volume_1,_Proposition_III.5.2_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu 1963</a>, Volume 1, Proposition III.5.2.</span> </li> <li id="cite_note-FOOTNOTEKobayashiNomizu1963Volume_1,_III.2-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKobayashiNomizu1963Volume_1,_III.2_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu 1963</a>, Volume 1, III.2.</span> </li> <li id="cite_note-FOOTNOTEKobayashiNomizu1963Volume_1,_III.5-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKobayashiNomizu1963Volume_1,_III.5_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu 1963</a>, Volume 1, III.5.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Hehl, F. W., & Obukhov, Y. N. (2007). <a rel="nofollow" class="external text" href="https://arxiv.org/abs/0711.1535">Elie Cartan's torsion in geometry and in field theory, an essay</a>. arXiv preprint arXiv:0711.1535.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFKobayashiNomizu1963">Kobayashi & Nomizu (1963)</a>, Chapter III, Section 4</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Bilby, B. A., Bullough, R., & Smith, E. (1955). <a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1955.0171">Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry</a>. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 231(1185), 263-273.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><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 class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Torsion">"Torsion"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Torsion&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTorsion&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+tensor" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Ozakin, A., & Yavari, A. (2014). Affine development of closed curves in Weitzenböck manifolds and the Burgers vector of dislocation mechanics. <a href="/wiki/Mathematics_and_Mechanics_of_Solids" class="mw-redirect" title="Mathematics and Mechanics of Solids">Mathematics and Mechanics of Solids</a>, 19(3), 299-307.</span> </li> <li id="cite_note-FOOTNOTEGorielyRobertson-TessiTaborVandiver2006-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGorielyRobertson-TessiTaborVandiver2006_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGorielyRobertson-TessiTaborVandiver2006">Goriely et al. 2006</a>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Trautman (1980) <i>Comments on the paper by Elie Cartan: Sur une generalisation de la notion de courbure de Riemann et les espaces a torsion</i>. In Bergmann, P. G., & De Sabbata, V. Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity (Vol. 58). Springer Science & Business Media.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">See Spivak (1999) Volume II, Addendum 1 to Chapter 6. See also Bishop and Goldberg (1980), section 5.10.</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=Torsion_tensor&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBishopGoldberg1980" class="citation cs2"><a href="/wiki/Richard_L._Bishop" title="Richard L. Bishop">Bishop, R.L.</a>; Goldberg, S.I. (1980), <i>Tensor analysis on manifolds</i>, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensor+analysis+on+manifolds&rft.pub=Dover+Publications&rft.date=1980&rft.aulast=Bishop&rft.aufirst=R.L.&rft.au=Goldberg%2C+S.I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+tensor" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartan1923" class="citation cs2"><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan, É.</a> (1923), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=ASENS_1923_3_40__325_0">"Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)"</a>, <i>Annales Scientifiques de l'École Normale Supérieure</i>, <b>40</b>: 325–412, <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/10.24033%2Fasens.751">10.24033/asens.751</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&rft.atitle=Sur+les+vari%C3%A9t%C3%A9s+%C3%A0+connexion+affine%2C+et+la+th%C3%A9orie+de+la+relativit%C3%A9+g%C3%A9n%C3%A9ralis%C3%A9e+%28premi%C3%A8re+partie%29&rft.volume=40&rft.pages=325-412&rft.date=1923&rft_id=info%3Adoi%2F10.24033%2Fasens.751&rft.aulast=Cartan&rft.aufirst=%C3%89.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASENS_1923_3_40__325_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+tensor" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartan1924" class="citation cs2"><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan, É.</a> (1924), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=ASENS_1924_3_41__1_0">"Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)"</a>, <i>Annales Scientifiques de l'École Normale Supérieure</i>, <b>41</b>: 1–25, <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/10.24033%2Fasens.753">10.24033/asens.753</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&rft.atitle=Sur+les+vari%C3%A9t%C3%A9s+%C3%A0+connexion+affine%2C+et+la+th%C3%A9orie+de+la+relativit%C3%A9+g%C3%A9n%C3%A9ralis%C3%A9e+%28premi%C3%A8re+partie%29+%28Suite%29&rft.volume=41&rft.pages=1-25&rft.date=1924&rft_id=info%3Adoi%2F10.24033%2Fasens.753&rft.aulast=Cartan&rft.aufirst=%C3%89.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASENS_1924_3_41__1_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+tensor" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElzanowskiEpstein1985" class="citation cs2">Elzanowski, M.; Epstein, M. 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Sciama">Sciama, D.W.</a> (1964), <a rel="nofollow" class="external text" href="http://rmp.aps.org/abstract/RMP/v36/i1/p463_1">"The physical structure of general relativity"</a>, <i>Rev. Mod. 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1999" class="citation cs2"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, M.</a> (1999), <i>A comprehensive introduction to differential geometry, Volume II</i>, Houston, Texas: Publish or Perish, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-914098-71-3" title="Special:BookSources/0-914098-71-3"><bdi>0-914098-71-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+comprehensive+introduction+to+differential+geometry%2C+Volume+II&rft.place=Houston%2C+Texas&rft.pub=Publish+or+Perish&rft.date=1999&rft.isbn=0-914098-71-3&rft.aulast=Spivak&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+tensor" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span 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title="Template talk:Curvature"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Curvature" title="Special:EditPage/Template:Curvature"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Various_notions_of_curvature_defined_in_differential_geometry" style="font-size:114%;margin:0 4em">Various notions of <a href="/wiki/Curvature" title="Curvature">curvature</a> defined in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">Differential geometry <br />of curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Torsion_of_a_curve" title="Torsion of a curve">Torsion of a curve</a></li> <li><a href="/wiki/Frenet%E2%80%93Serret_formulas" title="Frenet–Serret formulas">Frenet–Serret formulas</a></li> <li><a href="/wiki/Radius_of_curvature_(applications)" class="mw-redirect" title="Radius of curvature (applications)">Radius of curvature (applications)</a></li> <li><a href="/wiki/Affine_curvature" title="Affine curvature">Affine curvature</a></li> <li><a href="/wiki/Total_curvature" title="Total curvature">Total curvature</a></li> <li><a href="/wiki/Total_absolute_curvature" title="Total absolute curvature">Total absolute curvature</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">Differential geometry <br />of surfaces</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principal_curvature" title="Principal curvature">Principal curvatures</a></li> <li><a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a></li> <li><a href="/wiki/Mean_curvature" title="Mean curvature">Mean curvature</a></li> <li><a href="/wiki/Darboux_frame" title="Darboux frame">Darboux frame</a></li> <li><a href="/wiki/Gauss%E2%80%93Codazzi_equations" title="Gauss–Codazzi equations">Gauss–Codazzi equations</a></li> <li><a href="/wiki/First_fundamental_form" title="First fundamental form">First fundamental form</a></li> <li><a href="/wiki/Second_fundamental_form" title="Second fundamental form">Second fundamental form</a></li> <li><a href="/wiki/Third_fundamental_form" title="Third fundamental form">Third fundamental form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Curvature_of_Riemannian_manifolds" title="Curvature of Riemannian manifolds">Curvature of Riemannian manifolds</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Scalar_curvature" title="Scalar curvature">Scalar curvature</a></li> <li><a href="/wiki/Sectional_curvature" title="Sectional curvature">Sectional curvature</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Curvature of connections</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Curvature_form" title="Curvature form">Curvature form</a></li> <li><a class="mw-selflink selflink">Torsion tensor</a></li> <li><a href="/wiki/Cocurvature" title="Cocurvature">Cocurvature</a></li> <li><a href="/wiki/Holonomy" title="Holonomy">Holonomy</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tensors" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tensors" title="Template:Tensors"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tensors" title="Template talk:Tensors"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tensors" title="Special:EditPage/Template:Tensors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tensors" style="font-size:114%;margin:0 4em"><a href="/wiki/Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor<br />definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a class="mw-selflink selflink">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul> </div></td></tr></tbody></table></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" 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Torsion tensor - Wikipedia