Distance difference representations of Riemannian manifolds
Distance difference representations of Riemannian manifolds
复制标题
黎曼流形的距离差表示
作者：
S. Ivanov
设M是一个完全黎曼流形，$$Fsubset M$$ F∧M是一个具有非空内部的集合。对于每一个$$xin M$$ x∈M，设$$D_x$$ dx表示$$F imes F$$ F × F上的函数，定义为$$D_x(y,z)=d(x,y)-d(x,z)$$ dx (y, z) = D (x, y) - D (x, z)，其中D为M中的测地线距离。从M到$$F imes F$$ F × F上的连续函数空间的映射$$xmapsto D_x$$ x x x，用$${mathcal {D}}_F$$ D F表示，称为M的距离差分表示。最近由Lassas和Saksala介绍的这种表示方式，其动机是地球物理成像等。我们证明了距离差分表示$${mathcal {D}}_F$$ df是其像$${mathcal {D}}_F(M)$$ df (M)上的局部双lipschitz同纯，并且证明了对于每一个开集$$Usubset M$$ U∧M，集合$${mathcal {D}}_F(U)$$ df (U)唯一地决定了U上的黎曼度规。此外，如果M对其直径、曲率和注入半径有先验界，则由$${mathcal {D}}_F(M)$$ D F (M)确定M是稳定的。这扩展并加强了Lassas和Saksala之前的结果。
Let M be a complete Riemannian manifold and $$Fsubset M$$ F ⊂ M a set with a nonempty interior. For every $$xin M$$ x ∈ M , let $$D_x$$ D x denote the function on $$F imes F$$ 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 $$xmapsto D_x$$ x ↦ D x from M to the space of continuous functions on $$F imes F$$ 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 $$Usubset M$$ 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.