Simple closed geodesics in dimensions $$\ge 3$$

IF 1.4 3区数学 Q1 MATHEMATICS

Journal of Fixed Point Theory and Applications Pub Date : 2024-01-19 DOI:10.1007/s11784-023-01092-6

引用次数: 0

Abstract

We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold M of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number N(t) of geometrically distinct closed geodesics of length $\le t$ grows exponentially.

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尺寸为 $$\ge 3$$ 的简单闭合大地线

摘要我们证明，对于至少三维的紧凑可变流形 M 上的一般黎曼或可逆芬斯勒度量，所有闭合大地线都是简单且互不相交的。利用孔特雷拉斯的结果（Ann Math 2(172):761-808, 2010; in：Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729-1739, 2011）表明，对于紧凑且简单连接流形上的一般黎曼度量，所有闭大地线都是简单的，且长度为 $\le t$ 的几何上不同的闭大地线的数量 N(t) 呈指数增长。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Fixed Point Theory and Applications 数学-数学

CiteScore

3.10

自引率

5.60%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Fixed Point Theory and Applications (JFPTA) provides a publication forum for an important research in all disciplines in which the use of tools of fixed point theory plays an essential role. Research topics include but are not limited to: (i) New developments in fixed point theory as well as in related topological methods, in particular: Degree and fixed point index for various types of maps, Algebraic topology methods in the context of the Leray-Schauder theory, Lefschetz and Nielsen theories, Borsuk-Ulam type results, Vietoris fractions and fixed points for set-valued maps. (ii) Ramifications to global analysis, dynamical systems and symplectic topology, in particular: Degree and Conley Index in the study of non-linear phenomena, Lusternik-Schnirelmann and Morse theoretic methods, Floer Homology and Hamiltonian Systems, Elliptic complexes and the Atiyah-Bott fixed point theorem, Symplectic fixed point theorems and results related to the Arnold Conjecture. (iii) Significant applications in nonlinear analysis, mathematical economics and computation theory, in particular: Bifurcation theory and non-linear PDE-s, Convex analysis and variational inequalities, KKM-maps, theory of games and economics, Fixed point algorithms for computing fixed points. (iv) Contributions to important problems in geometry, fluid dynamics and mathematical physics, in particular: Global Riemannian geometry, Nonlinear problems in fluid mechanics.