Distance difference representations of Riemannian manifolds

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 Lassas and 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 Lassas and Saksala.