Multiple Closed Geodesics on Positively Curved Finsler Manifolds

Abstract In this paper, we prove that on every Finsler manifold {(M,F)} with reversibility λ and flag curvature K satisfying {(\frac{\lambda}{\lambda+1})^{2}<K\leq 1} , there exist {[\frac{\dim M+1}{2}]} closed geodesics. If the number of closed geodesics is finite, then there exist {[\frac{\dim M}{2}]} non-hyperbolic closed geodesics. Moreover, there are three closed geodesics on {(M,F)} satisfying the above pinching condition when {\dim M=3} .

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Existence of closed geodesics on positively curved Finsler manifolds

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385706001064 ◽

2007 ◽

Vol 27 (03) ◽

pp. 957 ◽

Cited By ~ 23

Author(s):

HANS-BERT RADEMACHER

Keyword(s):

Closed Geodesics ◽

Finsler Manifolds ◽

Positively Curved

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Projectively flat Finsler manifolds with infinite dimensional holonomy

Forum Mathematicum ◽

10.1515/forum-2012-0008 ◽

2015 ◽

Vol 27 (2) ◽

Cited By ~ 5

Author(s):

Zoltán Muzsnay ◽

Péter T. Nagy

Keyword(s):

Lie Group ◽

Holonomy Group ◽

Finsler Manifold ◽

Flag Curvature ◽

Compact Lie Group ◽

Finsler Manifolds ◽

Infinite Dimensional ◽

Constant Flag Curvature ◽

Projectively Flat

AbstractRecently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group cannot be a compact Lie group if the Finsler manifold of dimension >2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant–Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.

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Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

Advances in Geometry ◽

10.1515/advgeom-2021-0023 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Ming Xu

Keyword(s):

Lie Algebra ◽

Finsler Space ◽

Finsler Manifold ◽

Constant Speed ◽

Flag Curvature ◽

Finsler Metric ◽

Coset Space ◽

Finsler Spaces ◽

Negatively Curved ◽

Positively Curved

Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

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Closed Geodesics on Positively Curved Finsler 3-Spheres

Advanced Nonlinear Studies ◽

10.1515/ans-2015-5007 ◽

2016 ◽

Vol 16 (1) ◽

pp. 159-171

Author(s):

Huagui Duan ◽

Hui Liu

Keyword(s):

Three Dimensional ◽

Flag Curvature ◽

Dimensional Sphere ◽

Closed Geodesics ◽

The Third ◽

Positively Curved

AbstractIn [33], Wang proved that for every Finsler three-dimensional sphere ${(S^{3},F)}$ with reversibility λ and flag curvature K satisfying ${(\lambda/({1+\lambda}))^{2}<K\leq 1}$, there exist at least three distinct closed geodesics. In this paper, we prove that for every Finsler three-dimensional sphere $(S^{3},F)$ with reversibility λ and flag curvature K satisfying ${(9/4)(\lambda/(1+\lambda))^{2}<K\leq 1}$ with ${\lambda<2}$, if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.

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On almost contact Finsler structures

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501261 ◽

2020 ◽

Vol 17 (08) ◽

pp. 2050126

Author(s):

Tayebeh Tabatabaeifar ◽

Behzad Najafi ◽

Mehdi Rafie-Rad

Keyword(s):

Mild Condition ◽

Finsler Manifold ◽

Flag Curvature ◽

Finsler Manifolds ◽

Constant Flag Curvature ◽

Berwald Space ◽

Randers Metrics

We introduce almost contact and cosymplectic Finsler manifolds. Then, we characterize almost contact Randers metrics. It is proved that a cosymplectic Finsler manifold of constant flag curvature must have vanishing flag curvature. We prove that every cosymplectic Finsler manifold is a Landsberg space, under a mild condition. Finally, we show that a cosymplectic Finsler manifold is a Douglas space if and only if it is a Berwald space.

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A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021178 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Muhammad Hamid ◽

Wei Wang

Keyword(s):

Closed Geodesic ◽

Poincaré Map ◽

Finsler Manifold ◽

Flag Curvature ◽

Poincare Map ◽

Closed Geodesics ◽

Symmetric Property ◽

Geodesic Problem ◽

Common Index

<p style='text-indent:20px;'>In this paper, we prove a symmetric property for the indices for symplectic paths in the enhanced common index jump theorem (cf. Theorem 3.5 in [<xref ref-type="bibr" rid="b6">6</xref>]). As an application of this property, we prove that on every compact Finsler manifold <inline-formula><tex-math id="M1">\begin{document}$ (M, \, F) $\end{document}</tex-math></inline-formula> with reversibility <inline-formula><tex-math id="M2">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and flag curvature <inline-formula><tex-math id="M3">\begin{document}$ K $\end{document}</tex-math></inline-formula> satisfying <inline-formula><tex-math id="M4">\begin{document}$ \left(\frac{\lambda}{\lambda+1}\right)^2<K\le 1 $\end{document}</tex-math></inline-formula>, there exist two elliptic closed geodesics whose linearized Poincaré map has an eigenvalue of the form <inline-formula><tex-math id="M5">\begin{document}$ e^{\sqrt {-1}\theta} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \frac{\theta}{\pi}\notin{\bf Q} $\end{document}</tex-math></inline-formula> provided the number of closed geodesics on <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula> is finite.</p>

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Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

Journal of Geometric Analysis ◽

10.1007/s12220-021-00723-z ◽

2021 ◽

Author(s):

Tianyu Ma ◽

Vladimir S. Matveev ◽

Ilya Pavlyukevich

Keyword(s):

Brownian Motion ◽

Diffusion Process ◽

Random Walks ◽

Diffusion Processes ◽

Riemannian Metric ◽

Finsler Manifold ◽

Finsler Manifolds ◽

Bounded Geometry ◽

And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

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