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arXiv · 2108.05478

Detecting intrinsic global geometry of an obstacle via the layered scattering

Gabriel Katz
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Abstract

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 $\epsilon$-neighborhood $\mathsf T(K, \epsilon)$ of $K$ in $M$. The geodesics emanate from $\partial M$ and terminate there, after a number of reflections from the boundary $\partial \mathsf T(K, \epsilon)$. We use $\lceil \dim(K)/2\rceil$ many tubes $\{\mathsf T(K, \epsilon_j)\}_j$ 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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Gabriel Katz. 2021-08-12. Detecting intrinsic global geometry of an obstacle via the layered scattering. https://arxiv.org/abs/2108.05478

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