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.

中文翻译：

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黎曼流形的距离差表示

设 M 是一个完整的黎曼流形，而 $$F\subset M$$ F ⊂ M 是一个内部非空的集合。对于每一个 $$x\in M$$ x ∈ M ，令 $$D_x$$ D x 表示 $$F\times F$$ F × F 上由 $$D_x(y,z)=d( x,y)-d(x,z)$$ D x ( y , z ) = d ( x , y ) - d ( x , z ) 其中 d 是 M 中的测地线距离。映射 $$x\mapsto D_x$$ x ↦ D x 从 M 到 $$F\times F$$ F × F 上的连续函数空间，记为 $${\mathcal {D}}_F$$ DF ，称为 M 的距离差表示。最近由 Lassas 和 Saksala 引入的这种表示法是由地球物理成像等推动的。我们证明距离差表示 $${\mathcal {D}}_F$$ DF 是其图像 $${\mathcal {D}}_F(M)$$ DF ( M ) 上的局部双李普希茨同胚对于每个开集 $$U\subset M$$ U ⊂ M，集合 $${\mathcal {D}}_F(U)$$ DF ( U ) 唯一地确定了 U 上的黎曼度量。此外，如果 M 对其直径、曲率和注入半径有先验界限，则从 $${\mathcal {D}}_F(M)$$ DF ( M ) 确定 M 是稳定的。这扩展并加强了 Lassas 和 Saksala 的早期结果。