Beschreibung:
<jats:title>Abstract</jats:title><jats:p>In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected <jats:italic>n</jats:italic>-dimensional manifold of positive Ricci curvature <jats:inline-formula><jats:alternatives><jats:tex-math>$$\text {Ric}\ge n-1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mtext>Ric</mml:mtext>
<mml:mo>≥</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> has length <jats:inline-formula><jats:alternatives><jats:tex-math>$$\le n \pi .$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>≤</mml:mo>
<mml:mi>n</mml:mi>
<mml:mi>π</mml:mi>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> This improves the bound <jats:inline-formula><jats:alternatives><jats:tex-math>$$8\pi (n-1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mn>8</mml:mn>
<mml:mi>π</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo>-</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/2203.09492">arXiv:2203.09492</jats:ext-link>, 2022).</jats:p>