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Detecting intrinsic global geometry of an obstacle via the layered scattering

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.