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This is a Quantitative Study study.

September 24, 2025Open Access

Quadratic growth of geodesics on the two-sphere

Key Points

The number of prime closed geodesics grows quadratically with respect to length.
Key tools include Franks' theorem on periodic points and cylindrical contact homology theory.
Reversible Finsler metrics are the focus, expanding the understanding of geodesic behavior.
This research enhances the mathematical exploration of geometric structures on the two-sphere.

Abstract

We prove that for any reversible Finsler metric on S2, the number of prime closed geodesics grows quadratically with respect to length. The main tools are an improvement on Franks' theorem about the number of periodic points of area-preserving annulus maps, and the theory of cylindrical contact homology in the complement of a link.

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Cite This Study

Bernhard Albach (Thu,) studied this question.

synapsesocial.com/papers/68d6e16f8b2b6861e4c3ffee https://doi.org/https://doi.org/10.48550/arxiv.2508.00147

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