Distance difference representations of Riemannian manifolds

TL;DR

This paper studies a way to represent complete Riemannian manifolds using functions derived from distance differences, proving that this representation uniquely and stably encodes the manifold's geometry under certain bounds.

Contribution

It proves the distance difference representation is a locally bi-Lipschitz homeomorphism and that it uniquely determines the Riemannian metric on open sets, extending previous results.

Findings

The representation is a locally bi-Lipschitz homeomorphism.

Open sets are uniquely determined by their images under the representation.

The reconstruction is stable under bounds on diameter, curvature, and injectivity radius.

Abstract

Let $M$ be a complete Riemannian manifold and $F\subset M$ a set with a nonempty interior. For every $x\in M$, let $D_x$ denote the function on $F\times F$ defined by $D_x(y,z)=d(x,y)-d(x,z)$ where $d$ is the geodesic distance in $M$. The map $x\mapsto D_x$ from $M$ to the space of continuous functions on $F\times F$, denoted by $\mathcal D_F$, is called a distance difference representation of $M$. This representation, introduced recently by M. Lassas and T. Saksala, is motivated by geophysical imaging among other things. We prove that the distance difference representation $\mathcal D_F$ is a locally bi-Lipschitz homeomorphism onto its image $\mathcal D_F(M)$ and that for every open set $U\subset M$ the set $\mathcal D_F(U)$ uniquely determines the Riemannian metric on $U$. Furthermore the determination of $M$ from $\mathcal D_F(M)$ is stable if $M$ has a priori bounds on its diameter, curvature, and injectivity radius. This extends and strengthens earlier results by M. Lassas and T. Saksala.