[Paper Review] The index growth and multiplicity of closed geodesics

Huagui Duan, Yiming Long|arXiv (Cornell University)|Mar 18, 2010

Advanced Differential Geometry Research3 citations

TL;DR

This paper extends previous results on closed geodesics by analyzing the index growth of irrational closed geodesics on compact simply connected Finsler manifolds. Using Morse theory and index iteration formulas, the authors prove the existence of at least two distinct prime closed geodesics on every compact simply connected irreversible or reversible Finsler 4-manifold, generalizing earlier results for 3-manifolds and spheres.

ABSTRACT

In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in \cite{LoD1} on rational and in \cite{DuL1}, \cite{Rad4} and \cite{Rad5} on completely non-degenerate closed geodesics on spheres and $\CP^2$ to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.

Motivation & Objective

• To extend the multiplicity results for closed geodesics from rational to irrational closed geodesics on Finsler manifolds.
• To generalize prior results on spheres and complex projective spaces to all compact simply connected Finsler manifolds.
• To establish the existence of at least two distinct prime closed geodesics on 4-dimensional compact simply connected Finsler manifolds.
• To analyze the index growth properties of iterates of irrational closed geodesics using the Bott-type iteration formula and Morse index theory.
• To resolve the multiplicity problem in higher dimensions without requiring pinching or bumpy conditions.

Proposed method

• Classify prime closed geodesics into rational and irrational types based on the presence of irrational rotation matrices in their basic normal form decomposition.
• Apply the Bott-type iteration formula for the Morse index of iterates, using the formula $ i(c^m) = ext{Tr}(D au^m) $, where $ D au $ is the linearized Poincaré map.
• Use the identity $ rac{2T}{n} ext{Tr} ig( ext{sum over } m ext{ and } l_m ig) = T ilde{i}(c) B(d,h) $ to analyze the Betti number growth in the free loop space.
• Employ contradiction arguments by showing that the Betti numbers $ b_{k} $ must be even integers, leading to contradictions when they are odd (e.g., $ b_1 = 1 otin 2bN_0 $).
• Adapt the proof techniques from [LoD1] and [DuL1] to the irreversible and reversible Finsler cases, using symmetry and index monotonicity properties.
• Use the fact that $ ilde{i}(c) = ext{average index} $ and $ B(d,h) $ is a known topological invariant to derive global constraints on the number of closed geodesics.

Experimental results

Research questions

• RQ1 Can the multiplicity result of at least two distinct closed geodesics be extended from 3- to 4-dimensional compact simply connected Finsler manifolds?
• RQ2 How do the index growth properties of irrational closed geodesics differ from those of rational ones, and can they be used to derive global multiplicity results?
• RQ3 What constraints do the Betti numbers of the free loop space impose on the number of geometrically distinct closed geodesics?
• RQ4 Can the contradiction argument based on even Betti numbers be generalized to irreversible Finsler metrics?
• RQ5 Does the existence of only one geometrically distinct prime closed geodesic lead to a contradiction in the 4-dimensional case under index iteration?

Key findings

• The paper proves that every compact simply connected irreversible or reversible Finsler 4-manifold admits at least two distinct prime closed geodesics.
• The contradiction arises when assuming only one geometrically distinct prime closed geodesic: it leads to $ b_1 = 1 otin 2bN_0 $, violating the evenness of Betti numbers derived from index iteration.
• For irrational closed geodesics, the index growth pattern is analyzed via the Bott-type formula and the identity $ rac{2T}{n} ext{Tr}( ext{sum}) = T ilde{i}(c) B(4,1) $, which leads to a contradiction when only one geodesic exists.
• The result extends previous work on rational geodesics in 3-manifolds and on non-degenerate geodesics on spheres and $ bC P^2 $, now covering all compact simply connected Finsler manifolds.
• The proof holds for both reversible (including Riemannian) and irreversible Finsler metrics, with minor modifications to the argument in the reversible case.
• The key contradiction $ 1 = b_1 = M_1 otin 2bN_0 $ is derived in multiple cases (e.g., $ i(c) = 0,1,2 $), confirming the impossibility of a single closed geodesic.