LAUSR

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