scholarly journals Multiple Closed Geodesics on Positively Curved Finsler Manifolds

2019 ◽  
Vol 19 (3) ◽  
pp. 495-518 ◽  
Author(s):  
Wei Wang
Keyword(s):  
Finsler Manifold ◽  
Flag Curvature ◽  
Closed Geodesics ◽  
Finsler Manifolds ◽  
Positively Curved

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} .

scholarly journals Projectively flat Finsler manifolds with infinite dimensional holonomy

2015 ◽  
Vol 27 (2) ◽  
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.

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

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.

Closed Geodesics on Positively Curved Finsler 3-Spheres

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.

On almost contact Finsler structures

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.

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

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&lt;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>

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

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.

Closed Geodesics on Positively Curved Manifolds

1982 ◽  
Vol 116 (2) ◽  
pp. 213 ◽  
Author(s):  
W. Ballmann ◽  
G. Thorbergsson ◽  
W. Ziller
Keyword(s):  
Closed Geodesics ◽  
Curved Manifolds ◽  
Positively Curved

scholarly journals Closed geodesics on positively curved Finsler spheres

2008 ◽  
Vol 218 (5) ◽  
pp. 1566-1603 ◽  
Author(s):  
Wei Wang
Keyword(s):  
Closed Geodesics ◽  
Positively Curved
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