TY - JOUR

T1 - Distance difference representations of Riemannian manifolds

AU - Ivanov, Sergei

PY - 2020/8/1

Y1 - 2020/8/1

N2 - Let M be a complete Riemannian manifold and F⊂M a set with a nonempty interior. For every x∈M , let Dx denote the function on F×F defined by Dx(y,z)=d(x,y)−d(x,z) where d is the geodesic distance in M. The map x↦Dx from M to the space of continuous functions on F×F , denoted by DF , 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 DF is a locally bi-Lipschitz homeomorphism onto its image DF(M) and that for every open set U⊂M the set DF(U) uniquely determines the Riemannian metric on U. Furthermore the determination of M from DF(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.

AB - Let M be a complete Riemannian manifold and F⊂M a set with a nonempty interior. For every x∈M , let Dx denote the function on F×F defined by Dx(y,z)=d(x,y)−d(x,z) where d is the geodesic distance in M. The map x↦Dx from M to the space of continuous functions on F×F , denoted by DF , 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 DF is a locally bi-Lipschitz homeomorphism onto its image DF(M) and that for every open set U⊂M the set DF(U) uniquely determines the Riemannian metric on U. Furthermore the determination of M from DF(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.

KW - Distance functions

KW - inverse problems

KW - Inverse problems

UR - http://www.scopus.com/inward/record.url?scp=85074713366&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/distance-difference-representations-riemannian-manifolds

U2 - 10.1007/s10711-019-00491-9

DO - 10.1007/s10711-019-00491-9

M3 - Article

VL - 207

SP - 167

EP - 192

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -