@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:-TTBXXW26-C
  skos:prefLabel "Riemannian geometry"@en, "géométrie riemannienne"@fr ;
  a skos:Concept ;
  skos:narrower psr:-Z3DTDX39-D .

psr:-Z3DTDX39-D
  skos:prefLabel "Myers's theorem"@en, "théorème de Bonnet-Schoenberg-Myers"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Myers%27s_theorem>, <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Bonnet-Schoenberg-Myers> ;
  a skos:Concept ;
  dc:created "2023-07-19"^^xsd:date ;
  skos:broader psr:-TTBXXW26-C ;
  skos:definition """<b>Myers's theorem</b>, also known as the <b>Bonnet–Myers theorem</b>, is a celebrated, fundamental  theorem in the mathematical field of Riemannian geometry.  It was discovered by Sumner Byron Myers in 1941. It asserts the following:
<br/>
<br/><style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">Let <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)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo>,</mo>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (M,g)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e27d2e539fd0c3a9a7efab6257abd17de7fc57" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="(M,g)"></span> be a complete  Riemannian manifold of dimension <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 n}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> whose Ricci curvature satisfies <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 {Ric} (g)\\\\geq (n-1)k}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>Ric</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>≥<!-- ≥ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>k</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {Ric} (g)\\\\geq (n-1)k}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213c3fc0e3c62e010ebeb776e19b0cc1e511d6eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.832ex; height:2.843ex;" alt="{\\\\displaystyle \\\\operatorname {Ric} (g)\\\\geq (n-1)k}"></span> for some positive real number <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.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>k</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle k.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb6778a29f576eb23da1dbddffb73b2571359ac" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="k."></span> Then any two points of <i>M</i> can be joined by a geodesic segment of length at most <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="{\\	extstyle {\\rac {\\\\pi }{\\\\sqrt {k}}}{\\	ext{.}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="false" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mi>π<!-- π --></mi>
<br/>            <msqrt>
<br/>              <mi>k</mi>
<br/>            </msqrt>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtext>.</mtext>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\	extstyle {\\rac {\\\\pi }{\\\\sqrt {k}}}{\\	ext{.}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f538faafec3666848f64dfd9e074ea654809a1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:3.708ex; height:3.843ex;" alt="{\\	extstyle {\\rac {\\\\pi }{\\\\sqrt {k}}}{\\	ext{.}}}"></span></div>
<br/>In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the  Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the  sectional curvature. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Myers%27s_theorem">https://en.wikipedia.org/wiki/Myers%27s_theorem</a>)"""@en, """En géométrie riemannienne, le théorème de Bonnet-Schoenberg-Myers montre comment des contraintes locales sur une métrique riemannienne imposent des conditions globales sur la géométrie de la variété. Sa démonstration repose sur une utilisation classique de la formule de la variation seconde. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Bonnet-Schoenberg-Myers">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Bonnet-Schoenberg-Myers</a>)"""@fr ;
  skos:altLabel "Bonnet-Myers theorem"@en ;
  skos:inScheme psr: ;
  dc:modified "2023-08-24"^^xsd:date .