Detecting intrinsic global geometry of an obstacle via layered scattering

TL;DR

This paper introduces a layered scattering method using bubbling tubes to determine the intrinsic geometry of an obstacle within a domain, addressing the challenge of trapped geodesic trajectories.

Contribution

It proposes a novel layered scattering approach with bubbling tubes to extract global geometric invariants of obstacles, extending Weyl's classical tube volume theory.

Findings

Layered scattering with bubbling tubes detects intrinsic geometry.

Method avoids trapped trajectories by geometric modification.

Invariants reflect obstacle's volume and curvature.

Abstract

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 $\epsilon$-neighborhood $\mathsf T(K, \epsilon)$ 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 $\partial \mathsf T(K, \epsilon)$. 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 $\partial \mathsf T(K, \epsilon)$. To rule out such a possibility, we modify the geometry of a tube $\mathsf T(K, \epsilon)$ by building it from spherical bubbles. We need to use $\lceil \dim(K)/2\rceil$ many bubbling tubes $\{\mathsf T(K, \epsilon_j)\}_j$ 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 $\mathsf T(K, \epsilon)$ and their volumes.