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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Submersion</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 man eine <a href="Differenzierbarkeit#Differenzierbare_Abbildungen_zwischen_differenzierbaren_Mannigfaltigkeiten" title="Differenzierbarkeit">differenzierbare Abbildung</a> zwischen zwei <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbaren Mannigfaltigkeiten</a> als <b>Submersion</b>, falls ihr <a href="Tangentialraum#Die_Totalableitung_einer_Abbildung" title="Tangentialraum">Differential</a> an jeder Stelle <a href="Surjektivit%C3%A4t" class="mw-redirect" title="Surjektivität">surjektiv</a> ist. Eine spezielle Klasse von Submersionen sind die in der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> betrachteten <a href="Riemannsche_Submersion" title="Riemannsche Submersion">Riemannschen Submersionen</a>.
</p><p>Punkte, an denen das Differential nicht surjektiv ist, nennt man <a href="Kritischer_Punkt_(Mathematik)" title="Kritischer Punkt (Mathematik)"><i>kritisch</i></a> oder <i>singulär</i>.
</p><p>Ein wichtiges Beispiel für eine Submersion ist die Projektion <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}\to \mathbb {R} ^{m}\;;\;(x_{1};x_{2};\cdots ;x_{m};\cdots ;x_{n})\mapsto (x_{1};x_{2};\cdots ;x_{m})}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}\;;\;(x_{1};x_{2};\cdots ;x_{m};\cdots ;x_{n})\mapsto (x_{1};x_{2};\cdots ;x_{m})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdf8f6332a16a002e26ed018cd05b5e8c565d94f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.116ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}\;;\;(x_{1};x_{2};\cdots ;x_{m};\cdots ;x_{n})\mapsto (x_{1};x_{2};\cdots ;x_{m})}" loading="lazy"></span> für <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 n\geq m}">
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<annotation encoding="application/x-tex">{\displaystyle n\geq m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55d283f7f34d2e5d5aa08d7239f13f97f18c9bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle n\geq m}" loading="lazy"></span> auf die ersten <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}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">Koordinaten</a> im <a href="Euklidischer_Raum" title="Euklidischer Raum">Euklidischen Raum</a>. Tatsächlich lässt sich jede Submersion durch geeignete Wahl von <a href="Atlas_(Mathematik)" title="Atlas (Mathematik)">Karten</a> lokal in Form einer solchen Projektion darstellen.
</p><p>Ist der Zielraum die reelle Gerade <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} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>, so ist eine differenzierbare Funktion genau dann eine Submersion, wenn ihr Differential nirgendwo identisch verschwindet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Blätterungen_und_Faserbündel"><span id="Bl.C3.A4tterungen_und_Faserb.C3.BCndel"></span>Blätterungen und Faserbündel</h2></div>
<p>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\colon M\rightarrow B}">
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<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle f\colon M\rightarrow B}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4ad89280646640f9640616bb426ddb57f3e4f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.133ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow B}" loading="lazy"></span> eine Submersion ist, dann bilden die <a href="Niveaumenge" title="Niveaumenge">Niveaumengen</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 f^{-1}(b),b\in B}">
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<mi>B</mi>
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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(b),b\in B}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/770d27afef876d18cd28932868c8dd50415435e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.096ex; height:3.176ex;" alt="{\displaystyle f^{-1}(b),b\in B}" loading="lazy"></span> eine <a href="Bl%C3%A4tterung" title="Blätterung">Blätterung</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 M}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
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</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>. Das folgt aus dem <a href="Satz_von_der_impliziten_Funktion" title="Satz von der impliziten Funktion">Satz von der impliziten Funktion</a>.
</p><p>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 M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
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</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> kompakt 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 f\colon M\rightarrow B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
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<mi>B</mi>
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<annotation encoding="application/x-tex">{\displaystyle f\colon M\rightarrow B}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4ad89280646640f9640616bb426ddb57f3e4f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.133ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow B}" loading="lazy"></span> eine Submersion ist, 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\colon M\rightarrow f(M)}">
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle f\colon M\rightarrow f(M)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94d62c82cceb0896bf9a581598d4f7d1ca9e20f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.899ex; height:2.843ex;" alt="{\displaystyle f\colon M\rightarrow f(M)}" loading="lazy"></span> ein Faserbündel mit den Niveaumengen als Fasern. Das ist die Aussage des <a href="Satz_von_Ehresmann" title="Satz von Ehresmann">Satzes von Ehresmann</a>.
</p>
<p>Ein Beispiel einer Submersion, deren Niveaumengen eine Blätterung, aber kein Faserbündel bilden, ist
</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 f\colon \left[-1,1\right]\times {\mathbb {R} }\rightarrow {\mathbb {R} }}">
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<annotation encoding="application/x-tex">{\displaystyle f\colon \left[-1,1\right]\times {\mathbb {R} }\rightarrow {\mathbb {R} }}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d038a9a58e21c9d434db9577d29627e3948c2d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.584ex; height:2.843ex;" alt="{\displaystyle f\colon \left[-1,1\right]\times {\mathbb {R} }\rightarrow {\mathbb {R} }}" loading="lazy"></span></dd>
<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 f\left(x,y\right)=\left(x^{2}-1\right)e^{y}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<mo>(</mo>
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<mo>)</mo>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle f\left(x,y\right)=\left(x^{2}-1\right)e^{y}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff89c4896dc50136d7bc7fbfa65cf40079e1eb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.129ex; height:3.343ex;" alt="{\displaystyle f\left(x,y\right)=\left(x^{2}-1\right)e^{y}}" loading="lazy"></span>.</dd></dl>
<p>Das Bild rechts zeigt die Projektion dieser Blätterung auf <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[-1,1\right]\times S^{1}}">
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle \left[-1,1\right]\times S^{1}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd527f35358b1721ea8d8b943277353d0b5005ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.877ex; height:3.176ex;" alt="{\displaystyle \left[-1,1\right]\times S^{1}}" loading="lazy"></span>, wobei die Identifikation <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}=\mathbb {R} /\mathbb {Z} }">
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<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5a2fd5a106dcb71873fbc177f259971ecae897a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.065ex; height:3.176ex;" alt="{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }" loading="lazy"></span> benutzt wird.
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<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Immersion_(Mathematik)" title="Immersion (Mathematik)">Immersion</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>John M. Lee: <i>Introduction to Smooth Manifolds</i> (= <i>Graduate Texts in Mathematics</i> 218). Springer, New York NY u. a. 2003, ISBN 0-387-95448-1.</li>
<li>R. Abraham, Jerrold E. Marsden, T. Ratiu: <i>Manifolds, Tensor Analysis and Applications</i> (= <i>Applied Mathematical Sciences</i> 75). 2nd edition. Springer, New York NY u. a. 1988, ISBN 0-387-96790-7.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"></span></span></span><b><a href="https://de.wiktionary.org/wiki/Submersion" class="extiw external" title="wikt:Submersion">Wiktionary: Submersion</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div></div><!--htdig_noindex--><div><div class="zim-footer">
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