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<title>Transversalität</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Transversalität</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>In der <a href="Differentialtopologie" title="Differentialtopologie">Differentialtopologie</a> bezeichnet <b>Transversalität</b> einen Begriff, der die gegenseitige Lage zweier <a href="Untermannigfaltigkeit" title="Untermannigfaltigkeit">Untermannigfaltigkeiten</a> beschreibt. Transversalität beschreibt in gewissem Sinne das Gegenteil von Tangentialität und stellt den „Normalfall“ (siehe <a href="#Stabilität">Stabilität</a> und <a href="Transversalit%C3%A4tssatz" title="Transversalitätssatz">Transversalitätssatz</a>) dar.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Seien <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}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" 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 Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbare Mannigfaltigkeiten</a>, mit <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}">

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<annotation encoding="application/x-tex">{\displaystyle T_{x}Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2644caf37127a839b31fa4550f43533f9f0a0cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.303ex; height:2.509ex;" alt="{\displaystyle T_{x}Y}" loading="lazy"></span> wird der <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> am Punkt <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\in Y}">

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<annotation encoding="application/x-tex">{\displaystyle x\in Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3df349a03b91b999fa3741fb41bccb1a8023ee42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.944ex; height:2.176ex;" alt="{\displaystyle x\in Y}" loading="lazy"></span> und mit <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 d}">

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> das <a href="Totales_Differential" title="Totales Differential">totale Differential</a> bezeichnet.

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<ul><li>Sei weiterhin <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 f\colon X\rightarrow Y}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f986e95e93b70de25a0084daf075cb02c3ccae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow Y}" loading="lazy"></span> eine differenzierbare Abbildung 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 Z\subseteq Y}">

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<annotation encoding="application/x-tex">{\displaystyle Z\subseteq Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/152d29f7f7efef1db07515b61325626de530d57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.552ex; height:2.343ex;" alt="{\displaystyle Z\subseteq Y}" loading="lazy"></span> eine <a href="Untermannigfaltigkeit" title="Untermannigfaltigkeit">Untermannigfaltigkeit</a>. Die <b>Abbildung</b> <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 f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> heißt <b>transversal</b> zu <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 Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span>, falls gilt:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle T_{f(x)}Y=T_{f(x)}Z+d_{x}f(T_{x}X)\quad \forall \,x\in f^{-1}(Z)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a39434e47635926632c6c7faaafdf12b55e6ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.114ex; height:3.509ex;" alt="{\displaystyle T_{f(x)}Y=T_{f(x)}Z+d_{x}f(T_{x}X)\quad \forall \,x\in f^{-1}(Z)}" loading="lazy"></span></dd></dl>

<ul><li>Seien <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 W,Z\subseteq Y}">

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<annotation encoding="application/x-tex">{\displaystyle W,Z\subseteq Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/602d4ed9725e602c567d60fd133819d2dbdceab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.021ex; height:2.509ex;" alt="{\displaystyle W,Z\subseteq Y}" loading="lazy"></span> Untermannigfaltigkeiten. Die <b>Untermannigfaltigkeit</b> <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 W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> heißt <b>transversal</b> zu <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 Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span>, falls gilt:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle T_{x}Y=T_{x}Z+T_{x}W\quad \forall \,x\in W\cap Z}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c511dd799be051215dfbdf4248d6037bdb4ae8c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.289ex; height:2.509ex;" alt="{\displaystyle T_{x}Y=T_{x}Z+T_{x}W\quad \forall \,x\in W\cap Z}" loading="lazy"></span>.</dd>

<dd>Dies ist äquivalent dazu, dass die natürliche <a href="Inklusionsabbildung" title="Inklusionsabbildung">Inklusionsabbildung</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 i\colon W\hookrightarrow Y}">

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<annotation encoding="application/x-tex">{\displaystyle i\colon W\hookrightarrow Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06cf1c74cfe21e4691573dd84ced7efadedd3d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.952ex; height:2.176ex;" alt="{\displaystyle i\colon W\hookrightarrow Y}" loading="lazy"></span> transversal ist zu <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 Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Bemerkungen">Bemerkungen</h2></div>

<ul><li>Die Summe der Vektorräume ist im Allgemeinen keine direkte Summe.</li>

<li>Transversalität von Untermannigfaltigkeiten ist eine symmetrische Relation: <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{\mbox{ transversal zu }}Z\Longleftrightarrow Z{\mbox{ transversal zu }}X}">

<mtext> transversal zu </mtext>

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<mtext> transversal zu </mtext>

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<annotation encoding="application/x-tex">{\displaystyle X{\mbox{ transversal zu }}Z\Longleftrightarrow Z{\mbox{ transversal zu }}X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a0d6fd88f54f314da8e652ab8b9b980078f897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:43.23ex; height:2.176ex;" alt="{\displaystyle X{\mbox{ transversal zu }}Z\Longleftrightarrow Z{\mbox{ transversal zu }}X}" loading="lazy"></span>, man sagt deshalb auch „<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}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" 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 Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> schneiden sich transversal“.</li>

<li>Zwei disjunkte Untermannigfaltigkeiten schneiden sich immer transversal.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ^{2},\;t\mapsto (t,t^{2}+\varepsilon )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936d5bfa4e6563c8e3ce598c80df8d4458435b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.97ex; height:3.176ex;" alt="{\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ^{2},\;t\mapsto (t,t^{2}+\varepsilon )}" loading="lazy"></span> ist transversal zu <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 \mathbb {R} \times \left\{0\right\}\subset \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \left\{0\right\}\subset \mathbb {R} ^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f361c6056fcd9c1ab39f405aa6cb7d985bf043d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.837ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} \times \left\{0\right\}\subset \mathbb {R} ^{2}}" loading="lazy"></span> genau dann, wenn <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 \varepsilon \neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon \neq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b164106e88d10861e8dbc94194c20a96202efbde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.344ex; height:2.676ex;" alt="{\displaystyle \varepsilon \neq 0}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon =0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cbe653d369abd71fc0be98efbf59e9edc1bdfb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon =0}" loading="lazy"></span>: Im einzigen Schnittpunkt <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 (0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"></span> stimmen die Tangentialräume überein, ihre Summe ergibt nicht den ganzen Tangentialraum von <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 \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" 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} ^{2}}" loading="lazy"></span>.</li>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"></span>: Kein Schnittpunkt, also transversal.</li>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon <0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49e52eefdae802b4e1648d3bdeea48f927bd0d85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon <0}" loading="lazy"></span>: In den (beiden) Schnittpunkten ergibt die Summe der Tangentialräume der Untermannigfaltigkeiten den ganzen Tangentialraum.</li></ul></li>

<li>Zwei Geraden in <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 \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" 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} ^{2}}" loading="lazy"></span> schneiden sich genau dann transversal, wenn sie nicht identisch sind.</li>

<li>Zwei Geraden in <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 \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" 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}}" loading="lazy"></span> schneiden sich genau dann transversal, wenn sie sich nicht schneiden.</li>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}\times \left\{0\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c69e31a1233139b91df1b635bdde446b42c404c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.095ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{k}\times \left\{0\right\}}" 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 \left\{0\right\}\times \mathbb {R} ^{l}}">

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{0\right\}\times \mathbb {R} ^{l}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/857917ca4d568e5fc20860f18c0ef333920435c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.728ex; height:3.176ex;" alt="{\displaystyle \left\{0\right\}\times \mathbb {R} ^{l}}" loading="lazy"></span> in <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 \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> schneiden sich genau dann transversal, wenn <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 k+l\geq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+l\geq n}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/107f466c7e778892a85f6ed14a08dc0d13fb0492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.238ex; height:2.343ex;" alt="{\displaystyle k+l\geq n}" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<p>Die ursprüngliche Motivation für die Definition der Transversalität liegt in der Frage, wann das Urbild einer Untermannigfaltigkeit <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 Z\subseteq Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subseteq Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/152d29f7f7efef1db07515b61325626de530d57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.552ex; height:2.343ex;" alt="{\displaystyle Z\subseteq Y}" loading="lazy"></span> unter einer differenzierbaren Abbildungen <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 f\colon X\rightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f986e95e93b70de25a0084daf075cb02c3ccae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow Y}" loading="lazy"></span> wieder eine Untermannigfaltigkeit (von <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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"></span>) ist. Dies ist der Fall, wenn <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 f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> transversal ist zu <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 Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span>.

</p><p>Um dies zu zeigen, schreibt 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 Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> lokal als Niveaumenge einer differenzierbaren Abbildung <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 g\colon V\rightarrow \mathbb {R} ^{l},\;V\subseteq Y{\mbox{ offen}}}">

<msup>

</mrow>

</mrow>

</msup>

<mtext> offen</mtext>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon V\rightarrow \mathbb {R} ^{l},\;V\subseteq Y{\mbox{ offen}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad32e516565e15223ec8fbffd3e594e3f1f86efb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.782ex; height:3.009ex;" alt="{\displaystyle g\colon V\rightarrow \mathbb {R} ^{l},\;V\subseteq Y{\mbox{ offen}}}" loading="lazy"></span>, also <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 Z\cap V=g^{-1}(0)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\cap V=g^{-1}(0)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ae6bbcc20b816e4e432cc742726283a580c9873.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.571ex; height:3.176ex;" alt="{\displaystyle Z\cap V=g^{-1}(0)}" loading="lazy"></span>.

Die zu erfüllende Bedingung lautet nun: <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 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" 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 0}" loading="lazy"></span> ist <a href="Regul%C3%A4rer_Wert" title="Regulärer Wert">regulärer Wert</a> von <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 g\circ f\colon X\rightarrow \mathbb {R} ^{l}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f\colon X\rightarrow \mathbb {R} ^{l}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec6c0beedaddedfbadac37f80b679a2e24ca3ed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.618ex; height:3.009ex;" alt="{\displaystyle g\circ f\colon X\rightarrow \mathbb {R} ^{l}}" loading="lazy"></span>, das heißt, die Tangentialabbildung <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 d_{x}(g\circ f)\colon T_{x}X\rightarrow \mathbb {R} ^{l}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{x}(g\circ f)\colon T_{x}X\rightarrow \mathbb {R} ^{l}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b04325d1e15ad19d451dc4cb569a2aa3ceaedc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.339ex; height:3.176ex;" alt="{\displaystyle d_{x}(g\circ f)\colon T_{x}X\rightarrow \mathbb {R} ^{l}}" loading="lazy"></span> ist surjektiv für alle <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\in (g\circ f)^{-1}(0)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in (g\circ f)^{-1}(0)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea2454d3dd915010b0abd9d23333723097a38779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.873ex; height:3.176ex;" alt="{\displaystyle x\in (g\circ f)^{-1}(0)}" loading="lazy"></span>. Durch elementare Umformungen zeigt man, dass diese Bedingung äquivalent ist zu

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{f(x)}Y=T_{f(x)}Z+d_{x}f(T_{x}X)\quad \forall \,x\in f^{-1}(Z)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a39434e47635926632c6c7faaafdf12b55e6ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.114ex; height:3.509ex;" alt="{\displaystyle T_{f(x)}Y=T_{f(x)}Z+d_{x}f(T_{x}X)\quad \forall \,x\in f^{-1}(Z)}" loading="lazy"></span>, was der Definition der Transversalität entspricht.

</p><p>Die Umkehrung obiger Aussage trifft nicht zu. Dies sieht man wie folgt: 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 Y=S^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=S^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1065546567e100dcd80432940af915974739c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.448ex; height:2.676ex;" alt="{\displaystyle Y=S^{2}}" 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 X=S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=S^{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f682c320128f5f7fcb8bb8560f6e90813f9e0604.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.654ex; height:2.676ex;" alt="{\displaystyle X=S^{1}}" loading="lazy"></span> und seien <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 f_{i}\colon X\to Y}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}\colon X\to Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89059f7b6be379d0aacc8de2b7f6df8b79a050fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.34ex; height:2.509ex;" alt="{\displaystyle f_{i}\colon X\to Y}" loading="lazy"></span> Einbettungen der <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^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"></span> in Form geschlossener Kurven, die sich an einer Stelle tangential treffen (figure-eight). Fasse <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 \operatorname {Im} f_{1}=:Z}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} f_{1}=:Z}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25f90d24c6ee38f6bd79cbe151cf9645e026ac2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.782ex; height:2.509ex;" alt="{\displaystyle \operatorname {Im} f_{1}=:Z}" loading="lazy"></span> als eingebettete Untermannigfaltigkeit auf. Dann ist <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 f_{2}^{-1}(Z)=f_{2}^{-1}(\{*\})=\{*\}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}^{-1}(Z)=f_{2}^{-1}(\{*\})=\{*\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36e5a766870b86d365062c01827cced9fed02080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.777ex; height:3.343ex;" alt="{\displaystyle f_{2}^{-1}(Z)=f_{2}^{-1}(\{*\})=\{*\}}" loading="lazy"></span>, da <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 f_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc886fdaa7adc9be11ff4a5076da5e0943bcff58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{2}}" loading="lazy"></span> Einbettung.

</p>

<div class="mw-heading mw-heading2"><h2 id="Stabilität"><span id="Stabilit.C3.A4t"></span>Stabilität</h2></div>

<p>Eine Eigenschaft einer differenzierbarer Abbildungen <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\rightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\rightarrow Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d10a18ff36fb82a8fea00dc79971f5ab3a06caff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\rightarrow Y}" loading="lazy"></span> heißt <i>stabil</i>, wenn für

jede differenzierbare Homotopie <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 F\colon X\times [0,1]\rightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\colon X\times [0,1]\rightarrow Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b9b46459cb8eaa04a6f17f8bec4ad7d7c829786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.635ex; height:2.843ex;" alt="{\displaystyle F\colon X\times [0,1]\rightarrow Y}" loading="lazy"></span> gilt: Hat <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\mapsto F(x,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto F(x,0)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cdb222ea0a171aab45519fdcf5243033e79135f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.02ex; height:2.843ex;" alt="{\displaystyle x\mapsto F(x,0)}" loading="lazy"></span>

diese Eigenschaft, dann existiert ein <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 \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"></span> derart, dass <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\mapsto F(x,t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto F(x,t)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6696e8f9454d03c160b50f28816ae8b538a77f62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.697ex; height:2.843ex;" alt="{\displaystyle x\mapsto F(x,t)}" loading="lazy"></span>

diese Eigenschaft für alle <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\in [0,\varepsilon )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in [0,\varepsilon )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64f85fb7cc673dc1aa72b5c9e374338089e2acbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.512ex; height:2.843ex;" alt="{\displaystyle t\in [0,\varepsilon )}" loading="lazy"></span> auch besitzt.

</p><p>Der Stabilitätssatz besagt, dass für differenzierbare Abbildungen <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\rightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\rightarrow Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d10a18ff36fb82a8fea00dc79971f5ab3a06caff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\rightarrow Y}" loading="lazy"></span> die Transversalität zu einer abgeschlossenen Untermannigfaltigkeit <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 Z\subseteq Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subseteq Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/152d29f7f7efef1db07515b61325626de530d57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.552ex; height:2.343ex;" alt="{\displaystyle Z\subseteq Y}" loading="lazy"></span> eine stabile Eigenschaft ist, falls <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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"></span> <a href="Kompakter_Raum" title="Kompakter Raum">kompakt</a> ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Weitere_Sätze"><span id="Weitere_S.C3.A4tze"></span>Weitere Sätze</h2></div>

<p>Weitere wichtige Sätze in diesem Kontext sind der <a href="Transversalit%C3%A4tssatz" title="Transversalitätssatz">Transversalitätssatz</a> und der <a href="Transversalit%C3%A4tssatz#Relative_Version_und_Homotopietransversalit.C3.A4tssatz" title="Transversalitätssatz">Homotopietransversalitätssatz</a>. Sie besagen im Wesentlichen, dass zu jeder differenzierbaren Abbildungen eine homotope Abbildung existiert, welche zu einer vorgegebenen Untermannigfaltigkeit transversal ist und dass transversale Homotopien aus Familien von Abbildungen bestehen, welche für <a href="Fast_alle" title="Fast alle">fast alle</a> Parameterwerte transversal sind. Diese Sätze ermöglichen die allgemeine Definition von <a href="Schnittzahl" title="Schnittzahl">Schnittzahlen</a> mit Hilfe von Homotopie, da sich diese nur für transversale Schnitte direkt definieren lassen.

</p>

<div class="mw-heading mw-heading2"><h2 id="Generizität"><span id="Generizit.C3.A4t"></span>Generizität</h2></div>

<p>Eine Eigenschaft von Funktionen heißt <a href="Generische_Eigenschaft" title="Generische Eigenschaft">generisch</a>, wenn die Menge der Funktionen mit dieser Eigenschaft offen und dicht im Raum aller Funktionen ist.

</p><p>Transversalität (zu einer gegebenen Untermannigfaltigkeit <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 Z\subset Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\subset Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/742b3d8dd882c72e58909851bc707b35f08db486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;" alt="{\displaystyle Z\subset Y}" loading="lazy"></span>) ist eine generische Eigenschaft differenzierbarer Abbildungen <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\rightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\rightarrow Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d10a18ff36fb82a8fea00dc79971f5ab3a06caff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\rightarrow Y}" loading="lazy"></span>: Aus der <a href="#Stabilität">Stabilität</a> folgt die Offenheit und aus dem <a href="Transversalit%C3%A4tssatz" title="Transversalitätssatz">Transversalitätssatz</a> die Dichtheit der transversalen Abbildungen im Raum aller differenzierbaren Abbildungen.

</p>

<div class="mw-heading mw-heading2"><h2 id="Philosophie">Philosophie</h2></div>

<p>In der Philosophie wird der Begriff der Transversalität von <a href="Wolfgang_Welsch_(Philosoph)" title="Wolfgang Welsch (Philosoph)">Wolfgang Welsch</a> aufgegriffen.<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="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">W. Welsch: Vernunft. Die zeitgenössische Vernunftkritik und das Konzept der transversalen Vernunft. Frankfurt a. M. 1995.</span>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Victor_Guillemin" title="Victor Guillemin">Victor Guillemin</a>, Alan Pollack: <i>Differential topology.</i> Prentice-Hall, Englewood Cliffs NJ 1974, ISBN 0-13-212605-2.</li>

<li>Theodor Bröcker, Tammo tom Dieck: Kobordismentheorie, Lecture Notes in Mathematics 178, Springer Verlag (1970).</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-02-26" href="https://de.wikipedia.org/wiki/?title=Transversalit%C3%A4t&oldid=231287440">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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