@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-H0BLN01B-4
  skos:prefLabel "espace métrique"@fr, "metric space"@en ;
  a skos:Concept ;
  skos:narrower psr:-RPJ7R2LQ-0 .

psr: a skos:ConceptScheme .
psr:-TTBXXW26-C
  skos:prefLabel "Riemannian geometry"@en, "géométrie riemannienne"@fr ;
  a skos:Concept ;
  skos:narrower psr:-RPJ7R2LQ-0 .

psr:-RPJ7R2LQ-0
  skos:prefLabel "Cartan-Alexandrov-Toponogov space"@en, "espace de Cartan-Alexandrov-Toponogov"@fr ;
  skos:altLabel "CAT(k) space"@en, "espace CAT(k)"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Espace_de_Cartan-Alexandrov-Toponogov>, <https://en.wikipedia.org/wiki/CAT(k)_space> ;
  skos:definition """Les espaces de Cartan-Alexandrov-Toponogov ou espaces CAT(k) sont utilisés en géométrie. Ils permettent de définir dans le cadre des espaces métriques une notion de courbure qui relève traditionnellement de la géométrie riemannienne, par le truchement de relations de comparaison dans les triangles géodésiques. Le paramètre k est un réel qui permet de quantifier cette comparaison : on peut ainsi dire de certains espaces métriques qu'ils forment un espace CAT(k) pour un réel k donné. Les espaces CAT ont été dénommés ainsi par le géomètre Mikhail Gromov pour honorer les mathématiciens Élie Cartan, Alexandre Alexandrov et Victor Toponogov. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Espace_de_Cartan-Alexandrov-Toponogov">https://fr.wikipedia.org/wiki/Espace_de_Cartan-Alexandrov-Toponogov</a>)"""@fr, """In mathematics, a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {\\\\operatorname {\\	extbf {CAT}} } (k)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
         <mrow class="MJX-TeXAtom-ORD">
         <mtext mathvariant="bold">CAT</mtext>
         </mrow>
         </mrow>
         </mrow>
         <mo stretchy="false">(</mo>
         <mi>k</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {\\\\operatorname {\\	extbf {CAT}} } (k)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e48e739830947851ba1cbee11e1cbbe1bd9f99a8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.83ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {\\\\operatorname {\\	extbf {CAT}} } (k)}"></span> <b>space</b>, where <span class="mwe-math-element"><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}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>k</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> is a real number, is a specific type of metric space. Intuitively, triangles in a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\operatorname {CAT} (k)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>CAT</mi>
         <mo>⁡<!-- ⁡ --></mo>
         <mo stretchy="false">(</mo>
         <mi>k</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {CAT} (k)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2161800d8f9e6f312db6ae81e4a8e76e69c8ca" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.12ex; height:2.843ex;" alt="\\\\operatorname {CAT}(k)"></span> space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature <span class="mwe-math-element"><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}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>k</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>. In a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\operatorname {CAT} (k)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>CAT</mi>
         <mo>⁡<!-- ⁡ --></mo>
         <mo stretchy="false">(</mo>
         <mi>k</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {CAT} (k)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2161800d8f9e6f312db6ae81e4a8e76e69c8ca" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.12ex; height:2.843ex;" alt="\\\\operatorname {CAT}(k)"></span> space, the curvature is bounded from above by <span class="mwe-math-element"><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}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>k</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>. A notable special case is <span class="mwe-math-element"><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=0}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>k</mi>
         <mo>=</mo>
         <mn>0</mn>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle k=0}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="k=0"></span>; complete <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\operatorname {CAT} (0)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>CAT</mi>
         <mo>⁡<!-- ⁡ --></mo>
         <mo stretchy="false">(</mo>
         <mn>0</mn>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {CAT} (0)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77248875ff6817afa0812b0e71d6a80d12814b4a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.071ex; height:2.843ex;" alt="\\\\operatorname {CAT}(0)"></span> spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.
         
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/CAT(k)_space">https://en.wikipedia.org/wiki/CAT(k)_space</a>)"""@en ;
  skos:broader psr:-H0BLN01B-4, psr:-TTBXXW26-C ;
  skos:inScheme psr: ;
  a skos:Concept ;
  dc:created "2023-06-29"^^xsd:date ;
  dc:modified "2023-06-29"^^xsd:date .