[Paper Review] An inverse problem for compact Finsler manifolds with the boundary distance map

Maarten V. de Hoop, Joonas Ilmavirta+2 more|arXiv (Cornell University)|Jan 12, 2019

Advanced Differential Geometry Research35 references4 citations

TL;DR

This paper establishes that the boundary distance map uniquely determines the topological and differential structures of a smooth compact Finsler manifold with boundary. It identifies an optimal fiberwise open subset of the tangent bundle where the Finsler function is reconstructible from the boundary distance map, and proves unique determination when the Finsler function is fiberwise real analytic.

ABSTRACT

We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points.

Motivation & Objective

• To determine whether the boundary distance map encodes sufficient information to reconstruct the topological and differential structure of a compact Finsler manifold with boundary.
• To identify the largest subset of the tangent bundle where the Finsler function can be reconstructed from the boundary distance map.
• To establish conditions under which the Finsler function is uniquely determined by the boundary distance map.
• To analyze the smoothness properties of the distance function between interior and boundary points of the manifold.

Proposed method

• Constructing an optimal fiberwise open subset of the tangent bundle where the Finsler function is determined by the boundary distance map.
• Using the boundary distance map to infer geometric and topological invariants of the manifold.
• Applying fiberwise real analyticity assumptions to extend local reconstruction to global uniqueness of the Finsler function.
• Analyzing the regularity of the distance function from interior points to boundary points using inverse problem techniques.
• Employing differential geometric and inverse problem methods to relate boundary measurements to internal Finsler structure.
• Establishing that the Finsler function is not determined outside the identified optimal fiberwise open subset.

Experimental results

Research questions

• RQ1 Can the boundary distance map uniquely determine the topological and differential structure of a compact Finsler manifold with boundary?
• RQ2 What is the maximal subset of the tangent bundle where the Finsler function is reconstructible from the boundary distance map?
• RQ3 Under what conditions is the Finsler function uniquely determined by the boundary distance map?
• RQ4 How smooth is the distance function from interior points to boundary points in a Finsler manifold?
• RQ5 Is there a natural obstruction to reconstructing the Finsler function outside a specific fiberwise open subset of the tangent bundle?

Key findings

• The boundary distance map uniquely determines the topological and differential structures of the compact Finsler manifold.
• An optimal fiberwise open subset of the tangent bundle is identified where the Finsler function is determined by the boundary distance map.
• The Finsler function is not determined in the exterior of this optimal fiberwise open subset.
• If the Finsler function is fiberwise real analytic, it is uniquely determined by the boundary distance map.
• The distance function between interior and boundary points is shown to be smooth under the given geometric conditions.
• The construction of the optimal fiberwise open subset provides a sharp limit on the extent of reconstruction possible from boundary measurements.