Semiglobal boundary rigidity for Riemannian metrics
Semiglobal boundary rigidity for Riemannian metrics
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黎曼度量的半全局边界刚性
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
G. Uhlmann
中科院分区：
文献类型：
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作者：
M. Lassas;V. Sharafutdinov;G. Uhlmann
设（M，g）是一个有边黎曼流形.本文考虑由度量g的边界距离函数dg（x，y）= dist（x，y），x，y ∈ M，即边界点间的测地距离确定度量g的问题.这个问题出现在黎曼几何的刚性问题中[12]，[6]，[8]。对于M是欧几里得空间的有界域并且度量与欧几里得度量共形的情况，该问题被称为逆运动学问题，该问题起源于地球物理学，至少从20世纪初开始就有着悠久的历史。与Herglotz [10]一起的世纪。他考虑了M是球{x ∈ R| R =| X|其中c（r）是仅取决于半径r的正函数，|X|. Herglotz发现了一个根据边界距离函数确定c（r）的公式。物理上，这对应于折射率仅取决于半径的球对称地球模型的情况。边界距离函数对应于例如穿过地球并在表面处测量的声波的行进时间。声速依赖于所有变量的情况下的一般问题已经得到了广泛的研究（例如参见[17]和那里给出的参考文献）。此外，该问题与根据边界测量确定声速相关的其他逆问题有密切联系，参见[20]。
Let (M, g) be a Riemannian manifold with boundary. In this paper we consider the problem of determining the metric g from its associated boundary distance function dg(x, y) = dist (x, y), x, y ∈ ∂M, that is the geodesic distance between boundary points. This problem arose in rigidity questions in Riemannian geometry [12], [6], [8]. For the case in which M is a bounded domain of Euclidean space and the metric is conformal to the Euclidean one, this problem is called the inverse kinematic problem which arose in Geophysics and has a long history starting at least in the early part of the 20th century with Herglotz [10]. He considered the case where M is a ball {x ∈ R | r = |x| ≤ R} equipped with a spherically symmetric metric ds2 = dx2/c2(r) where c(r) is a positive function depending only on the radius r = |x|. Herglotz found a formula to determine c(r) from the boundary distance function. Physically this corresponds to the case of a spherically symmetric Earth model with an index of refraction depending only on the radius. The boundary distance function corresponds to the travel times of e.g. acoustic waves going through the Earth and measured at the surface. The general problem for the case that the sound speed depends on all variables has been extensively studied (see for instance [17] and the references given there). Also, this problem has a close connection for other inverse problems related to determining the sound speed from boundary measurements, see [20].