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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Riemannsche 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>Im <a href="Mathematik" title="Mathematik">mathematischen</a> Gebiet der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> bezeichnet man als <b>riemannsche Submersion</b> eine die <a href="Riemannsche_Metrik" class="mw-redirect" title="Riemannsche Metrik">riemannsche Metrik</a> respektierende <a href="Submersion" title="Submersion">Submersion</a> einer <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">riemannschen Mannigfaltigkeit</a> auf eine andere, die also lokal wie eine <a href="Orthogonale_Projektion" class="mw-redirect" title="Orthogonale Projektion">orthogonale Projektion</a> auf den <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> der zweiten Mannigfaltigkeit aussieht.
</p>
<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 (M,g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" 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 (N,h)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo>,</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (N,h)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04f1af6aecfdb83f8f264cb0631896fa969081a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.246ex; height:2.843ex;" alt="{\displaystyle (N,h)}" loading="lazy"></span> zwei riemannsche Mannigfaltigkeiten und
</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 M\to N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"></span></dd></dl>
<p>eine Submersion.
</p><p>Dann heißt <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</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> eine riemannsche Submersion, wenn der <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a>
</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 df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>f</mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">k</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">r</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>f</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>T</mi>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ea93c2a13af1849a7bedda679ce0ff7cb4ebbdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.828ex; height:3.176ex;" alt="{\displaystyle df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}" loading="lazy"></span></dd></dl>
<p>eine <a href="Isometrie_(Riemannsche_Geometrie)" title="Isometrie (Riemannsche Geometrie)">Isometrie</a> ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Konstruktion_von_Metriken_auf_Quotientenräumen"><span id="Konstruktion_von_Metriken_auf_Quotientenr.C3.A4umen"></span>Konstruktion von Metriken auf Quotientenräumen</h2></div>
<p>Eine <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</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 G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> <a href="Gruppenwirkung" class="mw-redirect" title="Gruppenwirkung">wirke</a> isometrisch, <a href="Freie_Wirkung" class="mw-redirect" title="Freie Wirkung">frei</a> und <a href="Eigentlich_diskontinuierlich" class="mw-redirect" title="Eigentlich diskontinuierlich">eigentlich diskontinuierlich</a> auf einer riemannschen Mannigfaltigkeit <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,g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" loading="lazy"></span>. Der <a href="Quotientenraum" class="mw-disambig" title="Quotientenraum">Quotientenraum</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 N=M/G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
<mo>=</mo>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N=M/G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b35e0fb8d7cd7be38e2fb6a7a1523eebb4aa2b35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.593ex; height:2.843ex;" alt="{\displaystyle N=M/G}" loading="lazy"></span> ist eine differenzierbare Mannigfaltigkeit und man hat einen Isomorphismus <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 df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>f</mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">k</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">r</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>f</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>T</mi>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ea93c2a13af1849a7bedda679ce0ff7cb4ebbdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.828ex; height:3.176ex;" alt="{\displaystyle df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}" loading="lazy"></span>.
</p><p>Eine Riemannsche Metrik 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 N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> wird eindeutig festgelegt durch, die Bedingung, dass dieser Isomorphismus eine <a href="Isometrie_(Riemannsche_Geometrie)" title="Isometrie (Riemannsche Geometrie)">Isometrie</a> sein soll. Sie wird als <b>Quotientenmetrik</b> bezeichnet. Mit dieser Metrik wird die Quotientenabbildung <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\to N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"></span> eine Riemannsche Submersion.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>Die <b>Fubini-Study-Metrik</b> auf dem <a href="Komplex-projektiver_Raum" class="mw-redirect" title="Komplex-projektiver Raum">komplex-projektiven Raum</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 \mathbb {C} P^{n}=S^{2n+1}/S^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<msup>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} P^{n}=S^{2n+1}/S^{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1873b15da24a522020349d82f2b240eeaf88df8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.217ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} P^{n}=S^{2n+1}/S^{1}}" loading="lazy"></span> ist die Quotientenmetrik für die Standard-Wirkung der <a href="Kreisgruppe" title="Kreisgruppe">Kreisgruppe</a> auf der „runden Sphäre“, also der Sphäre konstanter Schnittkrümmung +1. Mit dieser Metrik ist die Quotientenabbildung
</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 S^{2n+1}\to \mathbb {C} P^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<msup>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon S^{2n+1}\to \mathbb {C} P^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a51c71f5e09e8736e7d05a9baeb31179be89b46d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.307ex; height:3.009ex;" alt="{\displaystyle f\colon S^{2n+1}\to \mathbb {C} P^{n}}" loading="lazy"></span></dd></dl>
<p>also eine Riemannsche Submersion.
</p><p>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=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"></span> ist das die <a href="Hopf-Faserung" title="Hopf-Faserung">Hopf-Faserung</a> der Standardsphäre <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^{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01e57c690f890937838c10ba57853ff21bf30ec8.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^{3}}" loading="lazy"></span>: die Hopf-Abbildung
</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 H\colon S^{3}\to S^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>:<!-- : --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H\colon S^{3}\to S^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6396ea2b973181f85dd5effdee3d705379aeba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.863ex; height:2.676ex;" alt="{\displaystyle H\colon S^{3}\to S^{2}}" loading="lazy"></span></dd></dl>
<p>gibt eine Riemannsche Submersion.
</p>
<div class="mw-heading mw-heading2"><h2 id="O’Neill-Formel"><span id="O.E2.80.99Neill-Formel"></span>O’Neill-Formel</h2></div>
<p>Die <a href="Schnittkr%C3%BCmmung" title="Schnittkrümmung">Schnittkrümmung</a> des Bildraumes einer riemannschen Submersion kann aus der Schnittkrümmung des Urbildraumes mit der <b>O’Neill-Formel</b> berechnet werden:
</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 K_{N}(X,Y)=K_{M}({\tilde {X}},{\tilde {Y}})+{\tfrac {3}{4}}|[{\tilde {X}},{\tilde {Y}}]^{V}|^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>M</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>X</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>Y</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>X</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>Y</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msup>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{N}(X,Y)=K_{M}({\tilde {X}},{\tilde {Y}})+{\tfrac {3}{4}}|[{\tilde {X}},{\tilde {Y}}]^{V}|^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b130ba35e4da577eb5a577d30ce66f06c7ec2b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.312ex; height:3.676ex;" alt="{\displaystyle K_{N}(X,Y)=K_{M}({\tilde {X}},{\tilde {Y}})+{\tfrac {3}{4}}|[{\tilde {X}},{\tilde {Y}}]^{V}|^{2}}" loading="lazy"></span>.</dd></dl>
<p>Hierbei 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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy"></span> <a href="Orthonormal" class="mw-redirect" title="Orthonormal">orthonormale</a> <a href="Vektorfeld" title="Vektorfeld">Vektorfelder</a> 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 N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" 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 {\tilde {X}},{\tilde {Y}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>X</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>Y</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {X}},{\tilde {Y}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac5eff5f39abfe7ff47267e649b2cd7a4680d01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:3.009ex;" alt="{\displaystyle {\tilde {X}},{\tilde {Y}}}" loading="lazy"></span> ihre horizontalen Hochhebungen 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 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>, <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 [*,*]}">
<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 [*,*]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21a8c40ff6e68f423199145a4f4f3ef1f1f89cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [*,*]}" loading="lazy"></span> bezeichnet den <a href="Lie-Ableitung" title="Lie-Ableitung">Kommutator von Vektorfeldern</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 Z^{V}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>V</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z^{V}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1112f719796e3ed98c172bb2370ae2772cfef526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.204ex; height:2.676ex;" alt="{\displaystyle Z^{V}}" loading="lazy"></span> ist die Projektion des Vektorfeldes <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</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> auf die vertikale Distribution.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Jeff_Cheeger" title="Jeff Cheeger">Jeff Cheeger</a>, <a href="David_G._Ebin" class="mw-redirect" title="David G. Ebin">David G. Ebin</a>: <i>Comparison theorems in Riemannian geometry.</i> Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. ISBN 978-0-8218-4417-5</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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