Geodesic string counting invariants and arithmetic of multiplicities

2026/08/03 by Yasha Savelyev
Mathematics · #math.DG #math.DS #math.SG

paper · pdf

This supersedes arXiv:2309.09853, 45 pages, 1 figure

arxiv created 2026/08/03 · arxiv updated 2026/08/05

Abstract

We study rational valued counts of geodesic strings (reparametrization equivalence classes of closed geodesics) for complete Riemann-Finsler manifolds, based on the Fuller index of the geodesic flow. The main conceptual result is a product type formula for these counts. Combined with aspects of KAM theory, it yields the following sample phenomenon. Let g be a generic Finsler metric on T ², sufficiently C ^∞ -close to a flat metric, and fix a prime p and a nontrivial free homotopy class β. If there is a class β g-geodesic string with multiplicity divisible by p, then there is another one. We also obtain arithmetic constraints on counts of closed geodesics in mapping tori and flat bundles, and constraints on the existence of negative sectional curvature metrics. These counts can be understood as a shadow of a conjectural orbifold Morse homology of the infinite-dimensional quotient stack [LX/S¹].

Citations

• Extended Fuller index, sky catastrophes and the Seifert conjecture
• Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds
• Resonance for loop homology of spheres
• On a theorem due to Birkhoff
• A lecture on the classical KAM theorem
• A lecture on the classical KAM theorem
• On the energy functional on Finsler manifolds and applications to stationary spacetimes
• Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps
• An Introduction to Riemann-Finsler Geometry
• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• On the geometry and dynamics of diffeomorphisms of surfaces
• A fundamental class of geodesics on any closed surface of genus greater than one
• Untwisted Gromov-Witten invariants of Riemann-Finsler manifolds