Multiple Closed Geodesics on Positively Curved Finsler Manifolds

Wang, Wei

doi:10.1515/ans-2019-2043

Cited by 6 publications

(5 citation statements)

References 38 publications

(39 reference statements)

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“…In contrast to that result, we do not require the metric to be bumpy, but assume a topological condition on the manifold and a pinching condition on the flag curvature of the Finsler metric. An analogous statement that implies the existence of more than two positively distinct closed geodesics for Finsler metric on spheres under a pinching condition stricter than ours has been shown by Wang in [Wan19].…”

Section: Topological Complexity and Manifolds Dominated By Productssupporting

confidence: 83%

“…The same has been proven for reversible Finsler metrics on S 3 by Long and H. Duan in [LD09]. Recently, it has been shown by W. Wang in [Wan19] that if the flag curvature of a Finsler metric F on a closed manifold M satisfies the pinching condition λ 1+λ 2 < K ≤ 1, where λ denotes the reversibility of F, then (M, F) admits ⌊ dim M+1 2 ⌋ closed geodesics. As shown by Rademacher in [Rad04b], this pinching condition implies that M is homeomorphic to a sphere.…”

Section: Introductionmentioning

confidence: 54%

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Spherical complexities, with applications to closed geodesics

Mescher

2019

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We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik-Schnirelmann category and provide lower bounds for the numbers of critical orbits of SO(n)-invariant functions on spaces of n-spheres in a manifold. Lower bounds on these invariants are derived using weights of cohomology classes. As an application, we prove new existence results for closed geodesics on Finsler manifolds of positive flag curvature satisfying a pinching condition.Note that in contrast to the Morse-theoretic approach to critical points of a function which is often applied to energy functionals of Finsler metrics when studying closed geodesics, we do not demand any non-degeneracy condition on the critical points of F in Theorem 1.Lower bounds and cohomology. While sectional categories are in general hard to compute explicitly, there are several ways of obtaining lower and upper bounds on these numbers. Lower bounds are often derived in terms of the cohomology ring of the respective base space. More precisely, it was shown by Schwarz in [Sch66] that the sectional category of a fibration p : E → B is bounded from below by one plus the cup length of ker[p * : H * (B; R) → H * (E; R)] as an ideal in H * (B; R), where R is any commutative ring. Nevertheless, there are examples of fibrations for which the difference between this cup length and the actual value of secat(p) becomes arbitrarily big, hence its use is limited. An example is provided by the free loop space LS 2 of the 2-sphere. As shown in [FH89], the cup product of H * (LS 2 ; R) is trivial, while cat(LS 2 ) = +∞. Based on ideas of E. Fadell and S. Husseini for Lusternik-Schnirelmann category from [FH92] that were extended by Y. Rudyak in [Rud99], Farber and M. Grant introduced the notion of sectional category weight of a cohomology class with respect to a fibration p :one can associate a number wgt p (u i ) ∈ N with each of these classes, such that secat(p) will not only be bounded from below by n + 1, but by ∑ n i=1 wgt p (u i ) + 1. In particular, the mere existence of a cohomology class whose weight is at least k implies that secat(p) ≥ k + 1. In [GM18], Grant and the author have studied the condition on a cohomology class to have weight two or bigger in the case of the fibration defining topological complexity. The property of a cohomology class of having wgt p (u) ≥ 2 is equivalent to the property that u ∈ ker p * 2 , where p 2 : E 2 → B denotes the fiberwise join of p with itself. Fiberwise joins are given as homotopy pushouts of pullbacks, which always fit into a Mayer-Vietoris sequence, see [Str11, Chapter 21]. Grant and the author have studied the de Rham cohomology class of a symplectic form ω ∈ Ω 2 (M) on a closed even-dimensional manifold M in this context. They established a geometric criterion that allowed them to construct cohomology classes of weight two or bigger with respect to topological complexity out of the class [ω].In this articl...

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Section: Topological Complexity and Manifolds Dominated By Productssupporting

confidence: 83%

Section: Introductionmentioning

confidence: 54%

Spherical complexities, with applications to closed geodesics

Mescher

2019

Preprint

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“…Remark 4.5. The existence part of Corollary 4.4 is a consequence of a much stronger result by W. Wang from [Wan19], who has shown the existence of n distinct closed geodesics under the given assumptions. However, the methods used by Wang are entirely different than the ones used in this note and it does not follows from Wang's results that the length of the second-longest closed geodesic of F is less than 3ℓ F .…”

Section: Odd-dimensional Spheresmentioning

confidence: 84%

Existence results for closed Finsler geodesics via spherical complexities

Mescher

2020

Preprint

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We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths.

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“…The existence part of Corollary 4.4 is a consequence of a much stronger result by W. Wang from [20], who has shown the existence of n + 1 distinct closed geodesics under the weaker pinching condition λ 2 (1+λ) 2 < K ≤ 1. However, the methods used by Wang are entirely different than the ones used in this note and it does not follows from Wang's results that the length of the second-longest closed geodesic of F is less than 2 F .…”

Section: Remark 45mentioning

confidence: 86%