scholarly journals Two closed geodesics on compact simply connected bumpy Finsler manifolds

2016 ◽  
Vol 104 (2) ◽  
pp. 275-289 ◽  
Author(s):  
Huagui Duan ◽  
Yiming Long ◽  
Wei Wang
Keyword(s):  
Closed Geodesics ◽  
Finsler Manifolds ◽  
Simply Connected

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 Graphs and metric 2-step nilpotent Lie algebras

2018 ◽  
Vol 18 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Rachelle C. DeCoste ◽  
Lisa DeMeyer ◽  
Meera G. Mainkar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Inner Product ◽  
Nilpotent Lie Group ◽  
Closed Geodesics ◽  
Nilpotent Lie Algebra ◽  
Comprehensive Description ◽  
Nilpotent Lie Algebras ◽  
Simply Connected ◽  
Dense Set

AbstractDani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫G from a simple directed graph G in 2005. There is a natural inner product on 𝔫G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫g. We classify singularity properties of the Lie algebra 𝔫g in terms of the graph G. A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

THE BETTI NUMBERS OF THE FREE LOOP SPACE OF A CONNECTED SUM

2001 ◽  
Vol 64 (1) ◽  
pp. 205-228 ◽  
Author(s):  
PASCAL LAMBRECHTS
Keyword(s):  
Exponential Growth ◽  
Loop Space ◽  
Betti Numbers ◽  
Riemannian Metric ◽  
Closed Geodesics ◽  
Free Loop Space ◽  
Connected Sum ◽  
Simply Connected

Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the sequence (bk)k[ges ]1 has an exponential growth.Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic metric, the number of closed geodesics of length [les ]t grows exponentially with t.

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