AI 中文总结
该研究基于测地流的Fuller指标，推导测地弦计数的乘积型公式，结合KAM理论得出二维环面上测地弦重数的数论性质，还给出闭测地线计数及负截面曲率度量存在性的算术约束。
AI 中文摘要
我们基于测地流的Fuller指标，研究完备黎曼-芬斯勒流形上测地弦（闭测地线的重参数化等价类）的有理值计数。主要概念性结果是这些计数的乘积型公式。结合KAM理论的相关内容，它产生如下样本现象：设$g$为二维环面$T^2$上的一般芬斯勒度量，与平坦度量充分$C^\text{∞}$接近，固定素数$p$与非平凡自由同伦类$\beta$，若存在重数可被$p$整除的$\beta$型$g$-测地弦，则必存在另一个此类测地弦。我们还得到了映射环面与平坦丛中闭测地线计数的算术约束，以及负截面曲率度量存在性的约束。这些计数可被理解为无限维商栈$[LX/S^1]$的猜想性轨形莫尔斯同调的一个影子。
英文摘要
We study rational valued counts of geodesic strings (reparametrization equivalence classes of closed geodesics) for complete Riemann-Finsler manifolds, based on the Fuller index of the geodesic flow. The main conceptual result is a product type formula for these counts. Combined with aspects of KAM theory, it yields the following sample phenomenon. Let $g$ be a generic Finsler metric on $T ^{2}$, sufficiently $C ^{\infty }$-close to a flat metric, and fix a prime $p$ and a nontrivial free homotopy class $β$. If there is a class $β$ $g$-geodesic string with multiplicity divisible by $p$, then there is another one. We also obtain arithmetic constraints on counts of closed geodesics in mapping tori and flat bundles, and constraints on the existence of negative sectional curvature metrics. These counts can be understood as a shadow of a conjectural orbifold Morse homology of the infinite-dimensional quotient stack $[LX/S^1]$.
CommentsThis supersedes arXiv:2309.09853, 45 pages, 1 figure
