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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Torsionstensor</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Torsionstensor</b> ist ein mathematisches Objekt aus dem Bereich der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a>. Eingeführt wurde dieses Tensorfeld von <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> in seinen Studien zur Geometrie und <a href="Gravitation" title="Gravitation">Gravitation</a>.<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>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><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,\nabla )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (M,\nabla )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e56a4979d0171f5387171d7e367b46fe380e29b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.221ex; height:2.843ex;" alt="{\displaystyle (M,\nabla )}" loading="lazy"></span> eine <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbare Mannigfaltigkeit</a> zusammen mit einem <a href="Affiner_Zusammenhang" class="mw-redirect" title="Affiner Zusammenhang">affinen Zusammenhang</a> <span class="mwe-math-element mwe-math-element-inline"><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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"></span>. Der Torsionstensor <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> ist ein <a href="Tensorfeld" title="Tensorfeld">Tensorfeld</a>, das durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48c20ccb17431a4ccc168f73908ab0e3f22a98d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.837ex; height:2.843ex;" alt="{\displaystyle T(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]}" loading="lazy"></span></dd></dl>
<p>definiert ist. Dabei sind <span class="mwe-math-element mwe-math-element-inline"><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\in \Gamma (TM)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo>∈<!-- ∈ --></mo>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,Y\in \Gamma (TM)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbef8592dcceacd06769f997b10297decd8ac438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.968ex; height:2.843ex;" alt="{\displaystyle X,Y\in \Gamma (TM)}" loading="lazy"></span> zwei <a href="Vektorfeld" title="Vektorfeld">Vektorfelder</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\cdot ,\cdot ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\cdot ,\cdot ]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28dd4c22d60192519c1c12cf645b040f368db9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [\cdot ,\cdot ]}" loading="lazy"></span> stellt die <a href="Lie-Ableitung" title="Lie-Ableitung">Lie-Klammer</a> dar.<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>
<div class="mw-heading mw-heading2"><h2 id="Lokale_Darstellung">Lokale Darstellung</h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><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},\ldots ,e_{n}}">
<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>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}" loading="lazy"></span> ein lokaler <a href="Vektorb%C3%BCndel#Rahmen" title="Vektorbündel">Rahmen</a> des <a href="Tangentialb%C3%BCndel" title="Tangentialbündel">Tangentialbündels</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle TM}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea000afb5769206ddd5fd43f458430d04422ddeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}" loading="lazy"></span>. Das sind <a href="Schnitt_(Faserb%C3%BCndel)" title="Schnitt (Faserbündel)">Schnitte</a> im Tangentialbündel, die in jedem <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> eine <a href="Vektorraumbasis" class="mw-redirect" title="Vektorraumbasis">Vektorraumbasis</a> bilden. Setzt man <span class="mwe-math-element mwe-math-element-inline"><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:=e_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>:=</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X:=e_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/304a193a8053ce1375cf185712d64bf95db9224b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.608ex; height:2.509ex;" alt="{\displaystyle X:=e_{i}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y:=e_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>:=</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y:=e_{j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24bd3c16ccd5ffd561f4d227a5af1c9f2858fe6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.512ex; height:2.843ex;" alt="{\displaystyle Y:=e_{j}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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 _{ij}^{k}e_{k}:=[e_{i},e_{j}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>γ<!-- γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msubsup>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>:=</mo>
<mo stretchy="false">[</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 stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma _{ij}^{k}e_{k}:=[e_{i},e_{j}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fcb539e7eb1076f81700325450f5a36b53a76c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.803ex; height:3.509ex;" alt="{\displaystyle \gamma _{ij}^{k}e_{k}:=[e_{i},e_{j}]}" loading="lazy"></span>, dann gilt für die Komponenten <span class="mwe-math-element mwe-math-element-inline"><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_{ij}^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{ij}^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6dcb42295dc12e508f570492f7d912bd197eb0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.835ex; height:3.509ex;" alt="{\displaystyle T_{ij}^{k}}" loading="lazy"></span> des Torsionstensors in lokalen Koordinaten
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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<!-- 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<!-- 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<!-- 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"></mspace>
<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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07844fd720afbcaaaebf2d7b5acb9a456c716a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.021ex; 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.}" loading="lazy"></span></dd></dl>
<p>Dabei bezeichnen die Symbole <span class="mwe-math-element mwe-math-element-inline"><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 _{ij}^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma _{ij}^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54d6c913ed873bfa56bdf2e3dab2b3ea9225638a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.93ex; height:3.509ex;" alt="{\displaystyle \Gamma _{ij}^{k}}" loading="lazy"></span> die <a href="Christoffel-Symbol" class="mw-redirect" title="Christoffel-Symbol">Christoffel-Symbole</a>. Da es immer möglich ist, den lokalen Rahmen so zu wählen, dass die Lie-Klammer überall verschwindet, gilt in diesen Koordinaten für die Komponenten des Tensorfelds
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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},\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>
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</msub>
<mo>=</mo>
<msup>
<mi mathvariant="normal">Γ<!-- 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>
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</msub>
<mo>−<!-- − --></mo>
<msup>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msup>
<msub>
<mrow class="MJX-TeXAtom-ORD">
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="1em"></mspace>
<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},\quad i,j,k=1,2,\ldots ,n.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48ef301240b50c8d4c5ef36c06aa101d32bb2d37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.335ex; height:3.343ex;" alt="{\displaystyle T^{k}{}_{ij}=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji},\quad i,j,k=1,2,\ldots ,n.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<ul><li>Der Torsionstensor ist ein (2,1)-Tensorfeld, ist also insbesondere <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"></span>-<a href="Lineare_Abbildung" title="Lineare Abbildung">linear</a> in seinen drei Argumenten.</li>
<li>Der Torsionstensor ist schiefsymmetrisch, das heißt, es gilt <span class="mwe-math-element mwe-math-element-inline"><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)=-T(Y,X)}">
<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>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T(X,Y)=-T(Y,X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20afc61f8046d65d6bb57c796f83b49a5653872f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.372ex; height:2.843ex;" alt="{\displaystyle T(X,Y)=-T(Y,X)}" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Symmetrischer_Zusammenhang">Symmetrischer Zusammenhang</h2></div>
<p>Ein affiner Zusammenhang <span class="mwe-math-element mwe-math-element-inline"><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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"></span> heißt symmetrisch oder torsionsfrei, wenn der Torsionstensor verschwindet, wenn also
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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)=0}">
<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>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T(X,Y)=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1617e4be65972949e09873be29ef5d27c373d914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.494ex; height:2.843ex;" alt="{\displaystyle T(X,Y)=0}" loading="lazy"></span></dd></dl>
<p>oder äquivalent
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 _{X}Y-\nabla _{Y}X=[X,Y]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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 \nabla _{X}Y-\nabla _{Y}X=[X,Y]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c6e5703b32d7cbf681d8c5c7304eb681147024d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.763ex; height:2.843ex;" alt="{\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y]}" loading="lazy"></span></dd></dl>
<p>gilt. Der wichtigste symmetrische Zusammenhang ist der <a href="Levi-Civita-Zusammenhang" title="Levi-Civita-Zusammenhang">Levi-Civita-Zusammenhang</a>, der zusätzlich noch <a href="Metrischer_Zusammenhang" title="Metrischer Zusammenhang">metrisch</a> ist.
</p><p>Für symmetrische Zusammenhänge kann eine Art Verallgemeinerung des <a href="Satz_von_Schwarz" title="Satz von Schwarz">Satzes von Schwarz</a> für differenzierbare Kurven bewiesen werden. Sei <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> eine differenzierbare Mannigfaltigkeit mit symmetrischem Zusammenhang <span class="mwe-math-element mwe-math-element-inline"><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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∇<!-- ∇ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\colon \left]-\epsilon ,\epsilon \right[\times \left]a,b\right[\to M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>:<!-- : --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mo>−<!-- − --></mo>
<mi>ϵ<!-- ϵ --></mi>
<mo>,</mo>
<mi>ϵ<!-- ϵ --></mi>
</mrow>
<mo>[</mo>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
</mrow>
<mo>[</mo>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c\colon \left]-\epsilon ,\epsilon \right[\times \left]a,b\right[\to M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcbf6e199bd739787e95efdc8fd439c90b80b2bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.516ex; height:2.843ex;" alt="{\displaystyle c\colon \left]-\epsilon ,\epsilon \right[\times \left]a,b\right[\to M}" loading="lazy"></span> eine glatte <a href="Homotopie" title="Homotopie">Homotopie</a> von <a href="Glatte_Kurve" title="Glatte Kurve">glatten Kurven</a>, dann gilt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 _{\frac {\partial }{\partial s}}{\frac {\partial }{\partial t}}c(s,t)=\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial s}}c(s,t)\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>s</mi>
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</mfrac>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>s</mi>
</mrow>
</mfrac>
</mrow>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla _{\frac {\partial }{\partial s}}{\frac {\partial }{\partial t}}c(s,t)=\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial s}}c(s,t)\,.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56bdc65b3b198505a1d83d0eae7e80dc7796e986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.561ex; height:5.843ex;" alt="{\displaystyle \nabla _{\frac {\partial }{\partial s}}{\frac {\partial }{\partial t}}c(s,t)=\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial s}}c(s,t)\,.}" loading="lazy"></span></dd></dl>
<p>Einfach ausgedrückt kann im Fall eines symmetrischen Zusammenhangs also die Ableitung nach <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> mit der nach <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> vertauscht werden.<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>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><cite class="lang" lang="en" dir="auto" style="font-style:italic">Torsion tensor</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/torsion_tensor">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Torsionstensor&rft.atitle=Torsion+tensor&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Elie Cartan: <i>On manifolds with an Affine Connection and the Theory of General Relativity</i> (= <i>Monographs and Textbooks in Physical Science</i> 1). Bibliopolis, Neapol 1986, ISBN 88-7088-086-9 (Engl. transl. of French original 1922/23: <i>Sur les variétés à connexion affine et la théorie de la relativité généralisée</i>).</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">John M. Lee: <i>Riemannian Manifolds. An Introduction to Curvature</i> (= <i>Graduate Texts in Mathematics</i> 176). Springer, New York NY u. a. 1997, ISBN 0-387-98322-8, S. 68.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">John M. Lee: <i>Riemannian Manifolds. An Introduction to Curvature</i> (= <i>Graduate Texts in Mathematics</i> 176). Springer, New York NY u. a. 1997, ISBN 0-387-98322-8, S. 97–98.</span>
</li>
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