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Bumpy metrics on spheres and minimal index growth
Differential Geometry · Submitted 5 Aug 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27].
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The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this statement by the following observation: If for some $N \in \mathbb{N}$ all closed geodesics of index $\le N$ of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic $c$ then there is a covering $c^r$ of minimal index growth, i.e. $${\rm ind}(c^{rm})=m {\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \ge 1$ with ${\rm ind}\left(c^{rm}\right)\le N.$ But this leads to a contradiction for $N =\infty$ as pointed out by Goresky--Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large $L>0$ we obtain on $S^2$ a metric of positive flag curvature carrying only two closed geodesics of length $<L$ which do not intersect.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. | background 1.00 | 0.43 | 0.19 | asserted 1.00 |
| s1 | We simplify the proof of this statement by the following observation: If for some $N \in \mathbb{N}$ all closed geodesics of index $\le N$ of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic $c$ then there is a covering $c^r$ of minimal index growth, i.e. $${\rm ind}(c^{rm})=m {\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \ge 1$ with ${\rm ind}\left(c^{rm}\right)\le N.$ But this leads to a contradiction for $N =\infty$ as pointed out by Goresky--Hingston [13]. | contribution 0.47 | 0.86 | 0.32 | asserted 0.99 |
| s2 | We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. | contribution 0.97 | 0.71 | 0.05 | asserted 0.77 |
| s3 | For arbitrarily large $L>0$ we obtain on $S^2$ a metric of positive flag curvature carrying only two closed geodesics of length $<L$ which do not intersect. | result 0.99 | 0.89 | 0.23 | asserted 1.00 |
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Original abstract
The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this statement by the following observation: If for some $N \in \mathbb{N}$ all closed geodesics of index $\le N$ of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic $c$ then there is a covering $c^r$ of minimal index growth, i.e. $${\rm ind}(c^{rm})=m {\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \ge 1$ with ${\rm ind}\left(c^{rm}\right)\le N.$ But this leads to a contradiction for $N =\infty$ as pointed out by Goresky--Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large $L>0$ we obtain on $S^2$ a metric of positive flag curvature carrying only two closed geodesics of length $<L$ which do not intersect.
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States the same result 1 paper
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We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n>2.
The second closed geodesic on Finsler spheres of dimension n>2 · 1.00
What it cites
Cites as background 2 papers
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There are a number of results for metrics in higher dimensions with and without non-degeneracy assumptions or with curvature assumptions, cf. for example [ref] , [ref] , [ref] , [ref] , [ref] and [ref] .
The index quasi-periodicity and multiplicity of closed geodesics · Introduction · 0.97
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There are a number of results for metrics in higher dimensions with and without non-degeneracy assumptions or with curvature assumptions, cf. for example [ref] , [ref] , [ref] , [ref] , [ref] and [ref] .
The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds · Introduction · 0.97
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