Upper bounds for the critical values of homology classes of loops

2022/03/26 by Hans–Bert Rademacher, Rademacher, Hans-Bert
Mathematics · Physics and Astronomy · #53C22 #58E10 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.2203.14051

openalex publication_date 2022/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a simply-connected n-dimensional manifold of positive Ricci curvature \textrmRic ≥ n-1 has length ≤ n π.

Related

• Critical values of homology classes of loops and positive curvature
• Resonance for loop homology of spheres
• Free Loop Space Homology of Highly Connected Manifolds
• Positive Ricci Curvature and the Length of a shortest periodic geodesic
• Length spectrum of geodesic loops in manifolds of nonpositive curvature