scholarly journals Existence of closed geodesics on positively curved Finsler manifolds

2007 ◽  
Vol 27 (03) ◽  
pp. 957 ◽  
Author(s):  
HANS-BERT RADEMACHER
Keyword(s):  
Closed Geodesics ◽  
Finsler Manifolds ◽  
Positively Curved

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} .

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

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.

scholarly journals On interpolation and curvature via Wasserstein geodesics

2017 ◽  
Vol 10 (2) ◽  
pp. 125-167 ◽  
Author(s):  
Martin Kell
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Similar Condition ◽  
Interpolation Inequality ◽  
Metric Measure Spaces ◽  
Main Ingredient ◽  
Curvature Condition ◽  
Finsler Manifolds ◽  
Measure Spaces ◽  
Positively Curved

AbstractIn this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

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