基于几何的折纸定理及横向流动全息图一瞥
The Disk-Based Origami Theorem and a Glimpse of Holography for Traversing Flows
摘要：
本文描述了一种机制，通过这种机制，一个光滑连通流形$X$上的遍历泛流$v$产生一个紧$CW$ -复$\mathcal T(v)$，它同伦等价于$X$，并且使得$X$嵌入$\mathcal T(v)\times \mathbf R$ . $CW$ -复体$\mathcal T(v)$捕获关于$X$上的光滑结构的一些残余信息（例如$X$的稳定切丛）。此外，$\mathcal T(v)$是从一个单纯的\emph{origami map} $O: D^n \to \mathcal T(v)$，其源空间是一个磁盘$n = \dim(X) -1$的$D^n \subset \partial X$。$O$的纤维最多具有基数$(n+1)$。的知识的地图$O$，连同限制$v$的李雅普诺夫函数$f:X \to \mathbf R$的$D^n$，使得有可能重建的拓扑类型对$(X, \mathcal F(v))$，其中$\mathcal F(v)$是$1$ -叶，由$v$产生。这一事实促使在标题中使用“全息术”。
英文摘要：
This paper describes a mechanism by which a traversally generic flow $v$ on a smooth connected manifold $X$ with boundary produces a compact $CW$-complex $\mathcal T(v)$, which is homotopy equivalent to $X$ and such that $X$ embeds in $\mathcal T(v)\times \mathbf R$. The $CW$-complex $\mathcal T(v)$ captures some residual information about the smooth structure on $X$ (such as the stable tangent bundle of $X$). Moreover, $\mathcal T(v)$ is obtained from a simplicial \emph{origami map} $O: D^n \to \mathcal T(v)$, whose source space is a disk $D^n \subset \partial X$ of dimension $n = \dim(X) -1$. The fibers of $O$ have the cardinality $(n+1)$ at most. The knowledge of the map $O$, together with the restriction to $D^n$ of a Lyapunov function $f:X \to \mathbf R$ for $v$, make it possible to reconstruct the topological type of the pair $(X, \mathcal F(v))$, were $\mathcal F(v)$ is the $1$-foliation, generated by $v$. This fact motivates the use of "holography" in the title.
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