Stitching Data: Recovering a Manifold's Geometry from Geodesic Intersections

TL;DR

This paper introduces a method to recover the geometry of a Riemannian manifold with boundary using geodesic length and intersection data, termed 'stitching data', and applies it to a boundary measurement problem called the 'delayed collision data problem'.

Contribution

It establishes that geodesic intersection and length data uniquely determine the manifold's geometry, and demonstrates how to recover this geometry from collision data under certain conditions.

Findings

Geodesic intersection and length data suffice to recover manifold geometry.

The method applies to manifolds with a Riemannian collar of uniform radius.

Recovery is possible from collision data with geometric restrictions.

Abstract

Let $(M,g)$ be a Riemannian manifold with boundary. We show that knowledge of the length of each geodesic, and where pairwise intersections occur along the corresponding geodesics allows for recovery of the geometry of $(M,g)$ (assuming $(M,g)$ admits a Riemannian collar of a uniform radius). We call this knowledge the 'stitching data'. We then pose a boundary measurement type problem called the 'delayed collision data problem' and apply our first result about the stitching data to recover the geometry from the collision data (with some reasonable geometric restrictions on the manifold).