Lengths of three simple periodic geodesics on a Riemannian 2-sphere

2014/10/30 by Yevgeny Liokumovich, Liokumovich, Yevgeny, Alexander Nabutovsky +3 · 1 citation
Mathematics · #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1410.8456

openalex publication_date 2014/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a Riemannian 2-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on M. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed 20d, where d is the diameter of M. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bounds for these three lengths for "thin" spheres, when the area A is much less than d², where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal, when A\over d²\longrightarrow 0.

Cited by

• Short Simple Orthogonal Geodesic Chords on a 2-Disk with Convex Boundary

Related

• Short Simple Geodesic Loops on a 2-Sphere
• Lengths of geodesics between two points on a Riemannian manifold
• The length of the shortest closed geodesic on positively curved 2-spheres
• Positive Ricci Curvature and the Length of a shortest periodic geodesic
• Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area