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<title>Hyperbolischer Raum</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hyperbolischer Raum</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="Geometrie" title="Geometrie">Geometrie</a> ist der <b>hyperbolische Raum</b> ein Raum mit konstanter negativer Krümmung. Er erfüllt die Axiome der <a href="Euklidische_Geometrie" title="Euklidische Geometrie">euklidischen Geometrie</a> mit Ausnahme des <a href="Parallelenaxiom" title="Parallelenaxiom">Parallelenaxioms</a>. Der zweidimensionale hyperbolische Raum mit konstanter Krümmung <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 -1}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></span> heißt <a href="Hyperbolische_Ebene" title="Hyperbolische Ebene">hyperbolische Ebene</a>.

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

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> eine <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a>. Der <b>n-dimensionale hyperbolische Raum</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 \mathbb {H} ^{n}}">

<msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c802a2416834b80caf12cf130c97f085b4cfa9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {H} ^{n}}" loading="lazy"></span> ist die <a href="Dimension_(Mathematik)#Dimension_einer_Mannigfaltigkeit" title="Dimension (Mathematik)">n-dimensionale</a>,

<a href="Einfach_zusammenh%C3%A4ngend" class="mw-redirect" title="Einfach zusammenhängend">einfach zusammenhängende</a>, <a href="Geod%C3%A4tisch_vollst%C3%A4ndige_Mannigfaltigkeit" class="mw-redirect" title="Geodätisch vollständige Mannigfaltigkeit">vollständige</a> <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">Riemannsche Mannigfaltigkeit</a> mit <a href="Schnittkr%C3%BCmmung" title="Schnittkrümmung">Schnittkrümmung</a> konstant <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 -1}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></span>.

</p><p>Die Existenz des n-dimensionalen hyperbolischen Raumes ergibt sich aus den unten angegebenen Modellen, die Eindeutigkeit aus dem Satz von Cartan.

</p><p>Gelegentlich wird die Bezeichnung <i>hyperbolischer Raum</i> auch allgemeiner für <a href="Gromov-hyperbolischer_Raum" title="Gromov-hyperbolischer Raum"><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 \delta }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span>-hyperbolische Räume</a> im Sinne von <a href="Michail_Leonidowitsch_Gromow" title="Michail Leonidowitsch Gromow">Gromov</a> verwendet. Dieser Artikel betrachtet jedoch im Folgenden nur den hyperbolischen Raum mit Schnittkrümmung −1. Am Ende des Artikels werden weitere (teilweise nicht kompatible) in der Mathematik vorkommende Verwendungen des Begriffes „Hyperbolischer Raum“ aufgelistet.

</p>

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

<p>Aus einem <a href="Satz_von_Cartan-Ambrose-Hicks#Satz_von_Cartan" title="Satz von Cartan-Ambrose-Hicks">Satz von Elie Cartan</a> folgt, dass der n-dimensionale hyperbolische Raum bis auf Isometrie eindeutig ist. Insbesondere sind die unten angegebenen Modelle des n-dimensionalen hyperbolischen Raumes alle isometrisch zueinander.

</p>

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

<p>Zu jeder <a href="Geod%C3%A4te" title="Geodäte">Geodäte</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 L}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> und jedem 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 P\not \in L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\not \in L}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/850bb1bc69138df3611ede6341fd8554e28022bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.676ex;" alt="{\displaystyle P\not \in L}" loading="lazy"></span> gibt es unendlich viele 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 L}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> disjunkte Geodäten durch <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 P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>.

</p><p>Die <a href="Innenwinkelsumme" class="mw-redirect" title="Innenwinkelsumme">Innenwinkelsumme</a> von Dreiecken ist stets kleiner als <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 \pi }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span>. Der Flächeninhalt eines Dreiecks 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 \pi -(\alpha +\beta +\gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle \pi -(\alpha +\beta +\gamma )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd53da8d9455bcec9d7c891297266e0d89c1e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.744ex; height:2.843ex;" alt="{\displaystyle \pi -(\alpha +\beta +\gamma )}" loading="lazy"></span>, wobei <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 \alpha ,\beta ,\gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }" loading="lazy"></span> die Innenwinkel sind.

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<div class="mw-heading mw-heading3"><h3 id="Trigonometrie">Trigonometrie</h3></div>

<p>Es gelten die Formeln der hyperbolischen Trigonometrie:

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<mfrac>

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<mrow>

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</mfrac>

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<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin \alpha }{\sinh a}}={\frac {\sin \beta }{\sinh b}}={\frac {\sin \gamma }{\sinh c}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167ac85840e9dff98d01ebb529fac738db1062cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.545ex; height:5.509ex;" alt="{\displaystyle {\frac {\sin \alpha }{\sinh a}}={\frac {\sin \beta }{\sinh b}}={\frac {\sin \gamma }{\sinh c}}}" loading="lazy"></span></dd></dl>

<p>und

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cosh c=\cosh a\cosh b-\sinh a\sinh b\cos \gamma ,\,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39e24a9666beb972fcf5e6852cf4176b2be4a80c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.799ex; height:2.676ex;" alt="{\displaystyle \cosh c=\cosh a\cosh b-\sinh a\sinh b\cos \gamma ,\,}" loading="lazy"></span></dd></dl>

<p>wobei <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 \alpha ,\beta ,\gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }" loading="lazy"></span> die <a href="Innenwinkel" title="Innenwinkel">Innenwinkel</a> eines Dreiecks 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 a,b,c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> die Längen der gegenüberliegenden Seiten sind.

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<div class="mw-heading mw-heading3"><h3 id="Exponentielles_Wachstum">Exponentielles Wachstum</h3></div>

<p>Das Volumen eines Balles vom Radius <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 r}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> ist

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}\right)}}\int _{0}^{r}\sinh ^{n-1}(\rho )\,d\rho }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee07d854830bd8dc49c24ca6565b7c2bf340f897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.124ex; height:7.009ex;" alt="{\displaystyle {\frac {2\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}\right)}}\int _{0}^{r}\sinh ^{n-1}(\rho )\,d\rho }" loading="lazy"></span>,</dd></dl>

<p>es wächst somit exponentiell mit dem Radius.

</p>

<div class="mw-heading mw-heading3"><h3 id="Isometrien">Isometrien</h3></div>

<p>Geodätische Halbgeraden 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 {H} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c802a2416834b80caf12cf130c97f085b4cfa9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {H} ^{n}}" loading="lazy"></span> heißen <i>asymptotisch</i>, wenn sie endlichen Abstand haben. Dies definiert eine <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> auf der Menge der geodätischen Halbgeraden. Der <i><a href="Rand_im_Unendlichen" class="mw-redirect" title="Rand im Unendlichen">Rand im Unendlichen</a></i> <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 \partial _{\infty }\mathbb {H} ^{n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\infty }\mathbb {H} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7c38ec14bed0a3f0c52e737c765d9a3a5e5c2e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.136ex; height:2.676ex;" alt="{\displaystyle \partial _{\infty }\mathbb {H} ^{n}}" loading="lazy"></span> ist die Menge der Äquivalenzklassen von auf Bogenlänge parametrisierten geodätischen Halbgeraden. Jede Isometrie <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 \mathbb {H} ^{n}\rightarrow \mathbb {H} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {H} ^{n}\rightarrow \mathbb {H} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61ada26283998f8e5cc6fc2ccff4ee85460dad65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.98ex; height:2.676ex;" alt="{\displaystyle f\colon \mathbb {H} ^{n}\rightarrow \mathbb {H} ^{n}}" loading="lazy"></span> lässt sich auf den Rand im Unendlichen <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 \partial _{\infty }H^{n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\infty }H^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3cdf7518629e86cc44609ae1c7c842561b2c241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.432ex; height:2.676ex;" alt="{\displaystyle \partial _{\infty }H^{n}}" loading="lazy"></span> fortsetzen.

</p><p>Die Isometrien des hyperbolischen Raumes fallen in die folgenden (bis auf die Identitäts-Abbildung disjunkten) Klassen:

</p>

<ul><li><a href="Elliptische_Isometrie" title="Elliptische Isometrie"><i>elliptisch</i></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}">

</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> hat einen Fixpunkt 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 {H} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c802a2416834b80caf12cf130c97f085b4cfa9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {H} ^{n}}" loading="lazy"></span>,</li>

<li><a href="Loxodromische_Isometrie" class="mw-redirect" title="Loxodromische Isometrie"><i>loxodromisch</i></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}">

</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> hat keinen Fixpunkt 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 {H} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c802a2416834b80caf12cf130c97f085b4cfa9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {H} ^{n}}" loading="lazy"></span>, lässt aber zwei Punkte 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 \partial _{\infty }\mathbb {H} ^{n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\infty }\mathbb {H} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7c38ec14bed0a3f0c52e737c765d9a3a5e5c2e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.136ex; height:2.676ex;" alt="{\displaystyle \partial _{\infty }\mathbb {H} ^{n}}" loading="lazy"></span> und die sie verbindende Geodäte invariant,</li>

<li><a href="Parabolische_Isometrie" title="Parabolische Isometrie"><i>parabolisch</i></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}">

</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> lässt einen 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 p\in \partial _{\infty }\mathbb {H} ^{n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \partial _{\infty }\mathbb {H} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/877e21142d7edbbbf145b4fe3bb343b14d676815.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.236ex; height:2.676ex;" alt="{\displaystyle p\in \partial _{\infty }\mathbb {H} ^{n}}" loading="lazy"></span> und seine Horosphären invariant.</li></ul>

<p>Die Gruppe der Isometrien des <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 {H} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c802a2416834b80caf12cf130c97f085b4cfa9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {H} ^{n}}" loading="lazy"></span> ist isomorph 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 O^{+}(n,1)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O^{+}(n,1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdcca353d4b2a445a0cfa41ba414e2285979e559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.685ex; height:3.009ex;" alt="{\displaystyle O^{+}(n,1)}" loading="lazy"></span>.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Poincaré-Halbraum-Modell"><span id="Poincar.C3.A9-Halbraum-Modell"></span>Poincaré-Halbraum-Modell</h3></div>

<p>Der <a href="Halbraum" title="Halbraum">Halbraum</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 \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0\right\}\subset \mathbb {R} ^{n}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0\right\}\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b438bba7c87283ec104a77803c52d474fd8fe07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.723ex; height:2.843ex;" alt="{\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0\right\}\subset \mathbb {R} ^{n}}" loading="lazy"></span></dd></dl>

<p>mit der Riemannschen Metrik

</p>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {(dx_{1})^{2}+\ldots +(dx_{n})^{2}}{x_{n}^{2}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a7a1bd26dfd67a31fcb0eb1f362003f950189c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.331ex; height:6.509ex;" alt="{\displaystyle {\frac {(dx_{1})^{2}+\ldots +(dx_{n})^{2}}{x_{n}^{2}}}}" loading="lazy"></span></dd></dl>

<p>ist ein Modell des hyperbolischen Raumes.

</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=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.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=2}" loading="lazy"></span> wird es auch als Poincaré-Halbebenen-Modell bezeichnet.

</p>

<div class="mw-heading mw-heading3"><h3 id="Poincaré-Ball-Modell"><span id="Poincar.C3.A9-Ball-Modell"></span>Poincaré-Ball-Modell</h3></div>

<p>Die offene Kugel

</p>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+\ldots +x_{n}^{2}<1\right\}\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/006eb6ff0a6b4cac72a017ac29c6f0838ffebbfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.896ex; height:3.176ex;" alt="{\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+\ldots +x_{n}^{2}<1\right\}\subset \mathbb {R} ^{n}}" loading="lazy"></span></dd></dl>

<p>mit der Riemannschen Metrik

</p>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4{\frac {(dx_{1})^{2}+\ldots +(dx_{n})^{2}}{\left(1-x_{1}^{2}-\ldots -x_{n}^{2}\right)^{2}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8e2f4a38f610a1255d628fbd21a9029c1e689db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.494ex; height:7.509ex;" alt="{\displaystyle 4{\frac {(dx_{1})^{2}+\ldots +(dx_{n})^{2}}{\left(1-x_{1}^{2}-\ldots -x_{n}^{2}\right)^{2}}}}" loading="lazy"></span></dd></dl>

<p>ist ein Modell des hyperbolischen Raumes.

</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=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.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=2}" loading="lazy"></span> wird es auch als Poincaré-Kreisscheiben-Modell bezeichnet.

</p>

<div class="mw-heading mw-heading3"><h3 id="Hyperboloid-Modell">Hyperboloid-Modell</h3></div>

<p>Betrachte den <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+1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n+1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccea3976e1f8a1bb853c8ca00e52d518a3a4fe07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n+1}}" loading="lazy"></span> mit der <a href="Pseudo-riemannsche_Mannigfaltigkeit" title="Pseudo-riemannsche Mannigfaltigkeit">Pseudo-Riemannschen Metrik</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 -(dx_{1})^{2}+(dx_{2})^{2}+\ldots +(dx_{n+1})^{2}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -(dx_{1})^{2}+(dx_{2})^{2}+\ldots +(dx_{n+1})^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/619d007f0432b5e0fdb11b811ebdef1284dbba21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.707ex; height:3.176ex;" alt="{\displaystyle -(dx_{1})^{2}+(dx_{2})^{2}+\ldots +(dx_{n+1})^{2}}" loading="lazy"></span>.

</p><p>Das Hyperboloid

</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 \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {R} ^{n+1}:-x_{1}^{2}+x_{2}^{2}\ldots +x_{n+1}^{2}=-1,x_{1}>0\right\}\subset \mathbb {R} ^{n+1}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {R} ^{n+1}:-x_{1}^{2}+x_{2}^{2}\ldots +x_{n+1}^{2}=-1,x_{1}>0\right\}\subset \mathbb {R} ^{n+1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52b1537e439d55b98c3264d42bb8f2956803dce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:68.365ex; height:3.509ex;" alt="{\displaystyle \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {R} ^{n+1}:-x_{1}^{2}+x_{2}^{2}\ldots +x_{n+1}^{2}=-1,x_{1}>0\right\}\subset \mathbb {R} ^{n+1}}" loading="lazy"></span></dd></dl>

<p>mit der induzierten Metrik ist ein Modell des hyperbolischen Raumes.

</p>

<div class="mw-heading mw-heading3"><h3 id="Projektives_Modell">Projektives Modell</h3></div>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\colon \mathbb {R} ^{n+1}\setminus \{0\}\rightarrow \mathbb {R} P^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74c927257dac78f4e8d7fc67972c336b24d900f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.304ex; height:3.176ex;" alt="{\displaystyle p\colon \mathbb {R} ^{n+1}\setminus \{0\}\rightarrow \mathbb {R} P^{n}}" loading="lazy"></span> die kanonische Projektion auf den <a href="Projektiver_Raum" title="Projektiver Raum">projektiven Raum</a>, dann erhält man das projektive Modell des hyperbolischen Raumes als Bild des Hyperboloids unter <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 p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>.

</p><p>Nach der 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 \mathbb {R} P^{n}=\mathbb {R} ^{n}\cup \mathbb {R} P^{n-1}}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} P^{n}=\mathbb {R} ^{n}\cup \mathbb {R} P^{n-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f04dc5f35355ef55dd7a78598c2f9363f1aaa82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.114ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} P^{n}=\mathbb {R} ^{n}\cup \mathbb {R} P^{n-1}}" loading="lazy"></span> entspricht das projektive Modell der Menge

</p>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+\ldots +x_{n}^{2}<1\right\}\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/006eb6ff0a6b4cac72a017ac29c6f0838ffebbfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.896ex; height:3.176ex;" alt="{\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+\ldots +x_{n}^{2}<1\right\}\subset \mathbb {R} ^{n}}" loading="lazy"></span>.</dd></dl>

<p>Abstände berechnen sich gemäß der Hilbert-Metrik

</p>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

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</mrow>

<mrow>

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</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d(p,q)={\frac {1}{2}}\log {\frac {|qa||bp|}{|pa||bq|}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7ba8180c7af56446a9ce67c5f79ce0ca0505c1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.031ex; height:6.509ex;" alt="{\displaystyle d(p,q)={\frac {1}{2}}\log {\frac {|qa||bp|}{|pa||bq|}}}" loading="lazy"></span>,</dd></dl>

<p>wobei die Betragsstriche für euklidische Abstände stehen sollen 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 a,b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/181523deba732fda302fd176275a0739121d3bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}" loading="lazy"></span> die Schnittpunkte der Geodäten durch <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 p,q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953a97b9fe7d257c9666fb3cf6bf75380295e2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.362ex; height:2.009ex;" alt="{\displaystyle p,q}" loading="lazy"></span> mit der Einheitssphäre sind.

</p>

<div class="mw-heading mw-heading3"><h3 id="Historie">Historie</h3></div>

<p>Das Projektive Modell, das Poincaré-Ball-Modell und das Poincaré-Halbraum-Modell wurden 1868 von <a href="Eugenio_Beltrami" title="Eugenio Beltrami">Eugenio Beltrami</a> konstruiert, alle drei als Bilder eines weiteren (sogenannten „hemisphärischen“) Modells unter geeigneten Isometrien. Das Poincaré-Ball-Modell war 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=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.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=2}" loading="lazy"></span> bereits 1850 von <a href="Joseph_Liouville" title="Joseph Liouville">Liouville</a> untersucht worden und das projektive Modell kam 1859 in einer Arbeit <a href="Arthur_Cayley" title="Arthur Cayley">Cayleys</a> zur <a href="Projektive_Geometrie" title="Projektive Geometrie">projektiven Geometrie</a> vor, allerdings ohne Herstellung des Zusammenhangs zur hyperbolischen Geometrie.

</p><p>Zuvor hatten <a href="Nikolai_Iwanowitsch_Lobatschewski" title="Nikolai Iwanowitsch Lobatschewski">Nikolai Iwanowitsch Lobatschewski</a> und <a href="J%C3%A1nos_Bolyai" title="János Bolyai">János Bolyai</a> eine auf Axiomen aufbauende Theorie des hyperbolischen Raumes entwickelt und zahlreiche seiner Eigenschaften formal hergeleitet. Erst mit den von Beltrami angegebenen Modellen war aber der Beweis erbracht, dass die <a href="Hyperbolische_Geometrie" title="Hyperbolische Geometrie">hyperbolische Geometrie</a> widerspruchsfrei ist.

</p><p><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> entdeckte, dass die hyperbolische Geometrie auf natürliche Weise bei der Untersuchung von Differentialgleichungen und in der Zahlentheorie (bei der Untersuchung von <a href="Quadratische_Form" title="Quadratische Form">quadratischen Formen</a>) vorkommt. Im Zusammenhang mit der Untersuchung ternärer quadratischer Formen benutzte er 1881 erstmals das Hyperboloid-Modell.

</p>

<div class="mw-heading mw-heading2"><h2 id="Homogener_Raum">Homogener Raum</h2></div>

<p>Der hyperbolische Raum ist der <a href="Homogener_Raum" title="Homogener Raum">homogene Raum</a>

</p>

<msup>

</mrow>

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</msup>

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<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{n}=O(n,1)/(O(n)\times O(1))=O_{0}(n,1)/O(n)=SO_{0}(n,1)/SO(n),}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f2e83423776bfd2a791a9dad4d5e4e7361d4d8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.411ex; height:2.843ex;" alt="{\displaystyle \mathbb {H} ^{n}=O(n,1)/(O(n)\times O(1))=O_{0}(n,1)/O(n)=SO_{0}(n,1)/SO(n),}" loading="lazy"></span></dd></dl>

<p>wobei <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)O_{0}(n,1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (S)O_{0}(n,1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7390e36f44a07e8621dfd05c36ffbc9f0a6de38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.536ex; height:2.843ex;" alt="{\displaystyle (S)O_{0}(n,1)}" loading="lazy"></span> die <a href="Zusammenhangskomponente_der_Eins" title="Zusammenhangskomponente der Eins">Zusammenhangskomponente der Eins</a> 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 (S)O(n,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (S)O(n,1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac73cb7a811cbc19201879f48971afdeea1470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.482ex; height:2.843ex;" alt="{\displaystyle (S)O(n,1)}" loading="lazy"></span> bezeichnet.

</p><p>Damit ist hyperbolische Geometrie eine Geometrie im Sinne von Felix Kleins <a href="Erlanger_Programm" title="Erlanger Programm">Erlanger Programm</a>.

</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=2,3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2,3}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcd605706e3848b1c4f2a5690e6b4b010649b68a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.852ex; height:2.509ex;" alt="{\displaystyle n=2,3}" loading="lazy"></span> hat man auch die Darstellungen

</p>

<msup>

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</msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{2}=SL(2,\mathbb {R} )/SO(2)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccdb6da6eda7f7be689d1ab54bc84281af753e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.133ex; height:3.176ex;" alt="{\displaystyle \mathbb {H} ^{2}=SL(2,\mathbb {R} )/SO(2)}" loading="lazy"></span></dd>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ^{3}=SL(2,\mathbb {C} )/SU(2)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1be173717e268ecc63b61d6952dfe8959ff2925d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.143ex; height:3.176ex;" alt="{\displaystyle \mathbb {H} ^{3}=SL(2,\mathbb {C} )/SU(2)}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Einbettung_in_den_euklidischen_Raum">Einbettung in den euklidischen Raum</h2></div>

<p>Der hyperbolische Raum <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 {H} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c802a2416834b80caf12cf130c97f085b4cfa9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {H} ^{n}}" loading="lazy"></span> besitzt eine <a href="Isometrie_(Riemannsche_Geometrie)" title="Isometrie (Riemannsche Geometrie)">isometrische</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 C^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"></span>-<a href="Einbettung_(Mathematik)" title="Einbettung (Mathematik)">Einbettung</a> in den <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen 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 {R} ^{4n-3}}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4n-3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2d2e56234da73588afa94347fbc7b8ee0c0615b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.819ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4n-3}}" loading="lazy"></span>.<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="Andere_Verwendungen_des_Begriffs_„hyperbolischer_Raum“"><span id="Andere_Verwendungen_des_Begriffs_.E2.80.9Ehyperbolischer_Raum.E2.80.9C"></span>Andere Verwendungen des Begriffs „hyperbolischer Raum“</h2></div>

<ul><li>In der <a href="Metrischer_Raum" title="Metrischer Raum">metrischen Geometrie</a> 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 \delta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span>-hyperbolische Räume im Sinne von <a href="Michail_Leonidowitsch_Gromow" title="Michail Leonidowitsch Gromow">Gromov</a> (auch als Gromov-hyperbolische Räume bezeichnet) eine Klasse von metrischen Räumen, zu der unter anderem <a href="Einfach_zusammenh%C3%A4ngend" class="mw-redirect" title="Einfach zusammenhängend">einfach zusammenhängende</a> Mannigfaltigkeiten negativer Schnittkrümmung (insbesondere also auch der hyperbolische Raum) gehören. Endlich erzeugte Gruppen werden als <a href="Hyperbolische_Gruppe" title="Hyperbolische Gruppe">hyperbolische Gruppen</a> bezeichnet, wenn ihr <a href="Cayley-Graph" class="mw-redirect" title="Cayley-Graph">Cayley-Graph</a> 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 \delta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span>-hyperbolischer Raum ist.</li>

<li>In der Theorie der <a href="Symmetrischer_Raum" title="Symmetrischer Raum">symmetrischen Räume</a> gibt es neben den in diesem Artikel betrachteten hyperbolischen Räumen, die in diesem Zusammenhang oft als reell-hyperbolische Räume bezeichnet werden, noch die <a href="Komplex-hyperbolischer_Raum" title="Komplex-hyperbolischer Raum">komplex-hyperbolischen</a> und <a href="Quaternionisch-hyperbolischer_Raum" title="Quaternionisch-hyperbolischer Raum">quaternionisch-hyperbolischen Räume</a> sowie die Cayley-hyperbolische Ebene. Diese werden 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 \mathbb {K} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} }">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0bb0329ec640bc72c692a3c6b97f1f218a8cba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.139ex; height:2.509ex;" alt="{\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} }" loading="lazy"></span> oder <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 {O} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1ed2664a4fe515e6fbed25a7193ce663b82920c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }" loading="lazy"></span> definiert als <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\{(x_{1},\ldots ,x_{n+1})\in \mathbb {K} ^{n+1}:x_{1}^{2}-x_{2}^{2}\ldots -x_{n+1}^{2}=1\right\}\subset \mathbb {K} ^{n+1}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

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</mrow>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {K} ^{n+1}:x_{1}^{2}-x_{2}^{2}\ldots -x_{n+1}^{2}=1\right\}\subset \mathbb {K} ^{n+1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cba6b1f5a2e93aacd16787d16b47351d6b0a243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:57.33ex; height:3.509ex;" alt="{\displaystyle \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {K} ^{n+1}:x_{1}^{2}-x_{2}^{2}\ldots -x_{n+1}^{2}=1\right\}\subset \mathbb {K} ^{n+1}}" loading="lazy"></span> mit der induzierten Riemannschen Metrik.</li>

<li>In der <a href="Inzidenzgeometrie" title="Inzidenzgeometrie">Inzidenzgeometrie</a> ist ein hyperbolischer Raum ein angeordneter Inzidenzraum mit einer Kongruenzrelation und der Eigenschaft, dass jede Ebene mit der induzierten Anordnung und Kongruenzrelation eine hyperbolische Ebene im Sinne von Karzel-Sörensen-Windelberg<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> ist. Insbesondere gibt es in der endlichen Geometrie den Begriff endlicher hyperbolischer Räume.</li>

<li>In der <a href="Komplexe_Analysis" class="mw-redirect" title="Komplexe Analysis">komplexen Analysis</a> heißt eine <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexe Mannigfaltigkeit</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 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> Brody-hyperbolisch, wenn jede holomorphe 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 f\colon \mathbb {C} \rightarrow X}">

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</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ed0a3edbe91350ebf3c845ec7cc40963b59ca0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.585ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {C} \rightarrow X}" loading="lazy"></span> konstant ist. Dies gilt insbesondere für die durch das Poincaré-Kreisscheiben-Modell gegebene komplexe Struktur auf der hyperbolischen Ebene, siehe <a href="Satz_von_Liouville_(Funktionentheorie)" title="Satz von Liouville (Funktionentheorie)">Satz von Liouville</a>.</li>

<li>Ebenfalls in der komplexen Analysis heißt eine komplexe Mannigfaltigkeit Kobayashi-hyperbolisch (oder nur hyperbolisch), wenn die Kobayashi-Pseudo-Metrik eine Metrik ist. Für <a href="Kompakter_Raum" title="Kompakter Raum">kompakte</a> komplexe Mannigfaltigkeiten sind Brody-Hyperbolizität und Kobayashi-Hyperbolizität äquivalent.</li>

<li>In der komplexen Differentialgeometrie heißen <a href="K%C3%A4hler-Mannigfaltigkeit" title="Kähler-Mannigfaltigkeit">Kähler-Mannigfaltigkeiten</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 (M,\omega )}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d343da33a65bc6b0682b8da9ba31e5af966ce9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.731ex; height:2.843ex;" alt="{\displaystyle (M,\omega )}" loading="lazy"></span> <a href="K%C3%A4hler-hyperbolische_Mannigfaltigkeit" title="Kähler-hyperbolische Mannigfaltigkeit">Kähler-hyperbolisch</a>, wenn die hochgehobene <a href="K%C3%A4hlerform" class="mw-redirect" title="Kählerform">Kählerform</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 {\widetilde {\omega }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c35f59d8401f285eb56ddd649037d70532200807.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.03ex; margin-bottom: -0.308ex; width:1.446ex; height:2.176ex;" alt="{\displaystyle {\widetilde {\omega }}}" loading="lazy"></span> der <a href="Universelle_%C3%9Cberlagerung" class="mw-redirect" title="Universelle Überlagerung">universellen Überlagerung</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 {\widetilde {M}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/301b4dae6caafe7d093e43c068d75940062f7fe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.843ex;" alt="{\displaystyle {\widetilde {M}}}" loading="lazy"></span> das Differential einer beschränkten <a href="Differentialform" title="Differentialform">Differentialform</a> ist.</li>

<li>In der <a href="Homotopietheorie" class="mw-redirect" title="Homotopietheorie">Homotopietheorie</a> ist ein hyperbolischer Raum ein <a href="Topologischer_Raum" title="Topologischer Raum">topologischer 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 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> 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 \sum _{i}rk(\pi _{i}X)=\infty }">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i}rk(\pi _{i}X)=\infty }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdc4dc6d3081fb78f4e2915d546d7a9282401e01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.338ex; height:5.509ex;" alt="{\displaystyle \sum _{i}rk(\pi _{i}X)=\infty }" loading="lazy"></span>. Hier bezeichnet <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 \pi _{i}X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bc3520262c8eb08ca4c17d6ab286298d79dd284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.509ex;" alt="{\displaystyle \pi _{i}X}" loading="lazy"></span> die i-te <a href="Homotopiegruppe" title="Homotopiegruppe">Homotopiegruppe</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 rk}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ee5766411432122f625b195b7ab749a2e1fb40e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.26ex; height:2.176ex;" alt="{\displaystyle rk}" loading="lazy"></span> ihren <a href="Rang_einer_abelschen_Gruppe" title="Rang einer abelschen Gruppe">Rang</a>. Diese Definition steht in keinem Zusammenhang mit der in diesem Artikel besprochenen.</li></ul>

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

<ul><li>Eugenio Beltrami: <i>Saggio di interpretazione della geometria non-euclidea.</i> Giornale Matemat. 6 (1868), 284–312</li>

<li>Eugenio Beltrami: <i>Teoria fondamentale degli spazii di curvatura constante.</i> Ann. Mat. Ser. II 2 (1868–69), 232–255, <a href="https://doi.org/10.1007/BF02419615" class="extiw external" title="doi:10.1007/BF02419615">doi:10.1007/BF02419615</a>.</li>

<li>Felix Klein: <i>Über die sogenannte nicht-euklidische Geometrie</i> Math. Ann. 4 (1871), 573–625, <a href="https://doi.org/10.1007/BF01443189" class="extiw external" title="doi:10.1007/BF01443189">doi:10.1007/BF01443189</a>.</li>

<li>Henri Poincaré: <i>Théorie des groupes fuchsiens.</i> Acta Math. 1 (1882), 1–62 <a rel="nofollow" class="external text" href="http://www.univ-nancy2.fr/poincare/bhp/pdf/hp1882am.pdf">pdf</a></li>

<li>Henri Poincaré: <i>Mémoire sur les groupes kleinéens.</i> Acta Math. 3 (1883), 49–92 <a rel="nofollow" class="external text" href="http://www.univ-nancy2.fr/poincare/bhp/pdf/hp1883ama.pdf">pdf</a></li>

<li>Henri Poincaré: <i>Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques.</i> Assoc. Franç. Compt. Rend. 1881, 132–138 <a rel="nofollow" class="external text" href="http://www.univ-nancy2.fr/poincare/bhp/pdf/hp1881af.pdf">pdf</a></li></ul>

<p>Die 6 obigen Arbeiten sind ins Englische übersetzt in:

</p>

<ul><li>Stillwell, John: <i>Sources of hyperbolic geometry.</i> History of Mathematics, 10. American Mathematical Society, Providence, RI; London Mathematical Society, London, 1996. x+153 pp. ISBN 0-8218-0529-0</li></ul>

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

<ul><li>Cannon, Floyd, Kenyon, Parry: <a rel="nofollow" class="external text" href="http://www.math.brown.edu/~rkenyon/papers/cannon.pdf">Hyperbolic Geometry</a> (PDF; 425 kB)</li></ul>

<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">Oláh-Gál: <i>The n-dimensional hyperbolic space in E<sup>4n−3</sup>.</i> Publ. Math. Debrecen 46 (1995), no. 3-4, 205–213.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Karzel-Sörensen-Windelberg: <i>Einführung in die Geometrie.</i> Göttingen 1973</span>

</li>

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