基于分层散射的障碍物全局几何检测
Detecting intrinsic global geometry of an obstacle via layered scattering
摘要：
给定一个封闭的$k$维子流形$K$，incapsulated在一个紧凑的域$M \subset \mathbb E^n$，$k \leq n-2$，我们考虑的问题，确定的内在几何的障碍$K$（如体积，积分曲率）从散射数据，产生的反射测地线轨迹从边界的管状$ε$ -邻域$\mathsf T(K, ε)$的$K$在$M$。参与这种散射的测地线从边界$\partial M$发出，并在从边界$\partial \mathsf T(K, ε)$反射几次后终止于此。然而，这种设置的主要问题是，光线（台球轨迹）可能会在$K$附近卡住，因为它进入了那里的某个陷阱，所以这条光线会从$\partial \mathsf T(K, ε)$反射无限多次。为了排除这种可能性，我们修改了几何形状的管$\mathsf T(K, ε)$的建设从球形气泡。我们需要使用$\lceil \dim(K)/2\rceil$许多鼓泡管$\{\mathsf T(K, ε_j)\}_j$来检测$K$的某些全局不变量，这些不变量反映了它的内在几何形状。因此标题中的“分层散射”一词。赫尔曼·外尔在他的管的经典理论$\mathsf T(K, ε)$和它们的体积中研究了这些不变量。
英文摘要：
Given a closed $k$-dimensional submanifold $K$, incapsulated in a compact domain $M \subset \mathbb E^n$, $k \leq n-2$, we consider the problem of determining the intrinsic geometry of the obstacle $K$ (like volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular $ε$-neighborhood $\mathsf T(K, ε)$ of $K$ in $M$. The geodesics that participate in this scattering emanate from the boundary $\partial M$ and terminate there after a few reflections from the boundary $\partial \mathsf T(K, ε)$. However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of $K$ by entering some trap there so that this ray will have infinitely many reflections from $\partial \mathsf T(K, ε)$. To rule out such a possibility, we modify the geometry of a tube $\mathsf T(K, ε)$ by building it from spherical bubbles. We need to use $\lceil \dim(K)/2\rceil$ many bubbling tubes $\{\mathsf T(K, ε_j)\}_j$ for detecting certain global invariants of $K$, invariants which reflect its intrinsic geometry. Thus the words "layered scattering" in the title. These invariants were studied by Hermann Weyl in his classical theory of tubes $\mathsf T(K, ε)$ and their volumes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
