@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: a skos:ConceptScheme .
psr:-S78CS2MJ-M
  skos:prefLabel "variété riemannienne"@fr, "Riemannian manifold"@en ;
  a skos:Concept ;
  skos:narrower psr:-TRQCTHTV-Z .

psr:-TRQCTHTV-Z
  a skos:Concept ;
  skos:definition """In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold <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 (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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e27d2e539fd0c3a9a7efab6257abd17de7fc57" 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)}"></span> that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space <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 \\\\mathbb {R} ^{n}.}">
               <semantics>
               <mrow class="MJX-TeXAtom-ORD">
               <mstyle displaystyle="true" scriptlevel="0">
               <msup>
               <mrow class="MJX-TeXAtom-ORD">
               <mi mathvariant="double-struck">R</mi>
               </mrow>
               <mrow class="MJX-TeXAtom-ORD">
               <mi>n</mi>
               </mrow>
               </msup>
               <mo>.</mo>
               </mstyle>
               </mrow>
               <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n}.}</annotation>
               </semantics>
               </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {R} ^{n}.}"></span>. Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of <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 \\\\mathbb {R} ^{n}.}">
               <semantics>
               <mrow class="MJX-TeXAtom-ORD">
               <mstyle displaystyle="true" scriptlevel="0">
               <msup>
               <mrow class="MJX-TeXAtom-ORD">
               <mi mathvariant="double-struck">R</mi>
               </mrow>
               <mrow class="MJX-TeXAtom-ORD">
               <mi>n</mi>
               </mrow>
               </msup>
               <mo>.</mo>
               </mstyle>
               </mrow>
               <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n}.}</annotation>
               </semantics>
               </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {R} ^{n}.}"></span>. 
<br/> (Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Hadamard_manifold">https://en.wikipedia.org/wiki/Hadamard_manifold</a>)"""@en, """En géométrie riemannienne, une variété de Hadamard est une variété riemannienne complète, simplement connexe et de courbure sectionnelle négative ou nulle. En termes imagés, il s'agit d'une variété (sorte d'« espace courbe ») dans laquelle les géodésiques (analogues des « lignes droites ») issues d'un point donné, s'éloignent indéfiniment du point de départ et aussi les unes des autres. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Hadamard">https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Hadamard</a>)"""@fr ;
  skos:inScheme psr: ;
  skos:prefLabel "variété de Hadamard"@fr, "Hadamard manifold"@en ;
  dc:created "2023-08-31"^^xsd:date ;
  skos:related psr:-NG77K2NL-X ;
  skos:broader psr:-S78CS2MJ-M ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Hadamard_manifold>, <https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Hadamard> ;
  dc:modified "2024-10-18"^^xsd:date .

psr:-NG77K2NL-X
  skos:prefLabel "courbure négative"@fr, "negative curvature"@en ;
  a skos:Concept ;
  skos:related psr:-TRQCTHTV-Z .