Boundary and lens rigidity, tensor tomography and analytic microlocal analysis
Boundary and lens rigidity, tensor tomography and analytic microlocal analysis
复制标题
边界和晶状体刚性、张量断层扫描和微局部分析
DOI：
10.1007/978-4-431-73240-2_23
复制
发表时间：
2008
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
边界刚性问题包括确定一个紧凑的，黎曼流形的边界，直到等距，通过知道边界点之间的边界距离函数。透镜刚性包括确定的流形，通过知道的散射关系的措施，除了旅行时间，点和方向的出口的测地线从流形，如果一个知道它的点和方向的入口。张量层析成像是边界刚度和长度刚度的线性化。它包括确定一个对称张量的二阶从其积分沿着测地线。在本文中，我们调查这些问题上使用的方法从微观局部分析，特别是分析微观局部分析的一些最近的结果。虽然我们使用的是分布式版本的分析微局部分析，但许多想法都是基于佐藤学派的微局部分析的先驱工作，其中河合教授是非常重要的成员。
The boundary rigidity problem consists of determining a compact, Riemannian manifold with boundary, up to isometry, by knowing the boundary distance function between boundary points. Lens rigidity consists of determining the manifold, by knowing the scattering relation which measures, besides the travel times, the point and direction of exit of a geodesic from the manifold if one knows its point and direction of entrance. Tensor tomography is the linearization of boundary rigidity and length rigidity. It consists of determining a symmetric tensor of order two from its integral along geodesics. In this paper we survey some recent results obtained on these problems using methods from microlocal analysis, in particular analytic microlocal analysis. Although we use the distribution version of analytic microlocal analysis, many of the ideas were based on the pioneer work of the Sato school of microlocal analysis of which Professor Kawai was a very important member.