Distance difference representations of Riemannian manifolds

2018/06/13 by Ivanov, Sergei · 1 citation
#53C20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.05257

Abstract

Let M be a complete Riemannian manifold and F⊂ M a set with a nonempty interior. For every x∈ M, let D_x denote the function on F× F defined by D_x(y,z)=d(x,y)-d(x,z) where d is the geodesic distance in M. The map x↦ D_x from M to the space of continuous functions on F× 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⊂ 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.

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