通过分层散射检测障碍物的固有全局几何形状
Detecting intrinsic global geometry of an obstacle via the layered scattering
摘要：
给定一个紧致$k$维子流形$K \subset \mathbf R^n$，在一个紧致域$M \subset \mathbf R^n$中被封闭，我们考虑从$M$中$K$的管状$ε$ -邻域$\mathsf T(K, ε)$的边界反射测地线轨迹所产生的散射数据中确定障碍物$K$的内部几何的问题.测地线从$\partial M$发出，并在从边界$\partial \mathsf T(K, ε)$多次反射后终止于$\partial M$。我们使用$\lceil \dim(K)/2\rceil$许多管$\{\mathsf T(K, ε_j)\}_j$来检测$K$的某些全局内禀几何不变量，因此标题中的“分层散射”一词。赫尔曼·外尔在他的管理论中研究了这些不变量。
英文摘要：
Given a compact $k$-dimensional submanifold $K \subset \mathbf R^n$, incapsulated in a compact domain $M \subset \mathbf R^n$, we consider the problem of determining the inner geometry of the obstacle $K$ 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 emanate from $\partial M$ and terminate there, after a number of reflections from the boundary $\partial \mathsf T(K, ε)$. We use $\lceil \dim(K)/2\rceil$ many tubes $\{\mathsf T(K, ε_j)\}_j$ for detecting certain global intrinsic geometry invariants of $K$, thus the words "layered scattering" in the title. These invariants were studied by Hermann Weyl in his theory of tubes.
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