Detecting intrinsic global geometry of an obstacle via layered scattering

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.