Detecting intrinsic global geometry of an obstacle via the layered scattering

TL;DR

This paper investigates how to determine the intrinsic geometry of a submanifold obstacle within a domain by analyzing scattering data from geodesic reflections off layered tubular neighborhoods, extending Weyl's tube theory.

Contribution

It introduces a method to recover global geometric invariants of an obstacle using layered scattering data, based on multiple tubular neighborhoods.

Findings

Identifies geometric invariants from scattering data.

Uses multiple tubular layers for enhanced detection.

Extends Weyl's tube theory to scattering problems.

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.