Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds
Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds
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一类非简单黎曼流形的不完整数据的局部透镜刚度
DOI：
10.4310/jdg/1246888489
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发表时间：
2007
影响因子：
2.5
通讯作者：
G. Uhlmann
中科院分区：
文献类型：
--
作者：
Plamen Stefanov;G. Uhlmann
设\(\sigma\)是在具有不一定是凸边界的紧致黎曼流形\(M\)上的散射关系，它将边界上测地线射线的初始点和初始方向映射到边界上的出射点和出射方向。设\(\ell\)是该测地线射线的长度。我们研究这样一个问题：在知道\(\sigma\)以及限制在某个子集\(D\)上的\(\ell\)的情况下，度量\(g\)是否在等距意义下唯一确定。我们允许可能存在共轭点，但我们假设从\(D\)出发的测地线的余法丛覆盖\(T^{*}M\)；并且那些测地线没有共轭点。在一个额外的拓扑假设下，我们证明限制在\(D\)上的\(\sigma\)和\(\ell\)在一般度量附近局部唯一地恢复\(g\)的一个等距副本，特别是在实解析度量附近。
Let σ be the scattering relation on a compact Riemannian manifold M with nonnecessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let ` be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of σ and ` restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T ∗M ; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that σ and ` restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.