When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To ...When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.展开更多
Let(RP^(n),F)be a bumpy and irreversible Finsler n-dimensional real projective space with reversibilityλand flag curvature K satisfying(λ/1+λ)^(2)<K≤1 when n is odd,and K≥0 when n is even.We show that if there...Let(RP^(n),F)be a bumpy and irreversible Finsler n-dimensional real projective space with reversibilityλand flag curvature K satisfying(λ/1+λ)^(2)<K≤1 when n is odd,and K≥0 when n is even.We show that if there exist exactly 2[n+1/2]prime closed geodesics on such(RP^(n),F),then all of them are non-contractible,which coincides with the Katok’s examples.展开更多
In this paper we review and systematize the index method in closed geodesic problem.As we know,the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem,and has far from been resolved.I...In this paper we review and systematize the index method in closed geodesic problem.As we know,the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem,and has far from been resolved.In recent years,the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem,and has induced a great number of results on the multiplicity and stability of closed geodesics.We will systematically introduce these progresses in this review.展开更多
Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to ...Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form.In this paper,we prove that there are infinitely many non-contractible closed geodesics of class[h]on the compact space form with C^(r)-generic Finsler metrics,where 4≤r≤∞.The conclusion also holds for Cr-generic Riemannian metrics for 2≤r≤∞.The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.展开更多
基金Partially supported by NSF (10801079)Partially supported by RFDP (20080551002)+1 种基金Partially supported by LPMC of MOE of ChinaPartially supported by the 973 Program of MOST, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University
文摘In this paper, the concavity of closed geodesics proposed by M. Morse in 1930s is studied.
基金supported by National Natural Science Foundation of China(Grant Nos.11821101 and 11771331)Beijing Natural Science Foundation(Grant No.1182006)。
文摘When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.
基金The first author was partially supported by National Key R&D Program of China(Grant No.2020YFA0713300)NSFC(Grant Nos.11671215 and 11790271)+1 种基金LPMC of MOE of China and Nankai Universitythe second author was partially supported by NSFC(Grant Nos.11771341 and 12022111)。
文摘Let(RP^(n),F)be a bumpy and irreversible Finsler n-dimensional real projective space with reversibilityλand flag curvature K satisfying(λ/1+λ)^(2)<K≤1 when n is odd,and K≥0 when n is even.We show that if there exist exactly 2[n+1/2]prime closed geodesics on such(RP^(n),F),then all of them are non-contractible,which coincides with the Katok’s examples.
基金supported by Nankai Zhide Foundationsupported by NSFC(Grant Nos.12025101,11431001)。
文摘In this paper we review and systematize the index method in closed geodesic problem.As we know,the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem,and has far from been resolved.In recent years,the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem,and has induced a great number of results on the multiplicity and stability of closed geodesics.We will systematically introduce these progresses in this review.
基金supported by NSFC(Grant Nos.12371195,12022111)the Fundamental Research Funds for the Central Universities(Grant No.2042023kf0207)+1 种基金the second author was partially supported by NSFC(Grant No.11831009)Fundings of Innovating Activities in Science and Technology of Hubei Province。
文摘Let M=S^(n)/Γand h be a nontrivial element of finite order p inπ_(1)(Μ),where the integers n,p≥2,Γis a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form.In this paper,we prove that there are infinitely many non-contractible closed geodesics of class[h]on the compact space form with C^(r)-generic Finsler metrics,where 4≤r≤∞.The conclusion also holds for Cr-generic Riemannian metrics for 2≤r≤∞.The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.
