Detecting intrinsic global geometry of an obstacle via layered scattering

Abstract

Given a closed [Formula: see text]-dimensional submanifold [Formula: see text], encapsulated in a compact domain [Formula: see text], [Formula: see text], we consider the problem of determining the intrinsic geometry of the obstacle [Formula: see text] (such as volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular [Formula: see text]-neighborhood [Formula: see text] of [Formula: see text] in [Formula: see text]. The geodesics that participate in this scattering emanate from the boundary [Formula: see text] and terminate there after a few reflections from the boundary [Formula: see text]. However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of [Formula: see text] by entering some trap there so that this ray will have infinitely many reflections from [Formula: see text]. To rule out such a possibility, we modify the geometry of a tube [Formula: see text] by building it from spherical bubbles. We need to use [Formula: see text] many bubbling tubes [Formula: see text] for detecting certain global invariants of [Formula: see text], invariants that reflect its intrinsic geometry. Thus, the words “layered scattering” are in the title. These invariants were studied by Hermann Weyl in his classical theory of tubes [Formula: see text] and their volumes.