Abstract
AbstractIn 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 n-dimensional manifold of positive Ricci curvature $$\text {Ric}\ge n-1$$
Ric
≥
n
-
1
has length $$\le n \pi .$$
≤
n
π
.
This improves the bound $$8\pi (n-1)$$
8
π
(
n
-
1
)
given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).
Publisher
Springer Science and Business Media LLC
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