Detecting intrinsic global geometry of an obstacle via layered scattering.

Abstract

Given a closed k-dimensional submanifold K, encapsulated in a compact domain M ⊂ E, k ≤ n - 2, we consider the problem of determining the intrinsic geometry of the obstacle K (such as volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular ϵ-neighborhood T ( K , ϵ ) of K in M. The geodesics that participate in this scattering emanate from the boundary ∂ M and terminate there after a few reflections from the boundary ∂ T ( K , ϵ ). 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 ∂ T ( K , ϵ ). To rule out such a possibility, we modify the geometry of a tube T ( K , ϵ ) by building it from spherical bubbles. We need to use ⌈ dim ⁡ ( K ) / 2 ⌉ many bubbling tubes { T ( K , ϵ ) } for detecting certain global invariants of K, 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 T ( K , ϵ ) and their volumes.