Boundary rigidity and stability for generic simple metrics
Boundary rigidity and stability for generic simple metrics
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DOI：
10.1090/s0894-0347-05-00494-7
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发表时间：
2004-08
影响因子：
3.9
通讯作者：
Plamen Stefanov;G. Uhlmann
中科院分区：
文献类型：
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作者：
Plamen Stefanov;G. Uhlmann
设（M, g）是一个有边界的黎曼流形。用pg表示度规g中的距离函数。我们考虑pg（x,y）是否对所有x已知，y在<9M上是否唯一地决定度规的逆问题。这个问题出现在地球物理学中，试图通过测量地震波的传播时间来确定地球的内部结构。这可以追溯到赫格洛茨[H]、维切特和佐普里茨[WZ]。虽然重点一直在介质是各向同性的情况下，各向异性的情况已经引起了地球物理学的兴趣，因为它已经发现地球的内核表现出各向异性的行为[Cr]。在微分几何中，由于刚性问题而研究了这种反问题，称为边界刚性问题。很明显，不能唯一地确定度量。任何在边界处相等的等距都会得到相同的测量值。此外，边界距离函数只考虑了最短路径，很容易找到唯一判定的反例，因此需要对度量进行一定的限制。Michel [Mi]推测，一个简单度量g是唯一确定的，直到一个固定边界的微分同态作用，由dM上所有x和y已知的边界距离函数pg（x,y）决定。我们回顾定义1.1。我们说黎曼度规g在M中是简单的，如果dM是严格凸w.r.t. g，对于任意x g M，指数映射exp^: exp~1(M) ？M是一个微分同构。
Let (M, g) be a Riemannian manifold with boundary. Denote by pg the distance function in the metric g. We consider the inverse problem of whether pg(x,y), known for all x, y on <9M, determines the metric uniquely. This problem arose in geophysics in an attempt to determine the inner structure of the Earth by measuring the travel times of seismic waves. It goes back to Herglotz [H] and Wiechert and Zoeppritz [WZ]. Although the emphasis has been in the case that the medium is isotropic, the anisotropic case has been of interest in geophysics since it has been found that the inner core of the Earth exhibits anisotropic behavior [Cr]. In differential geometry this inverse problem has been studied because of rigidity questions and is known as the boundary rigidity problem. It is clear that one cannot determine the metric uniquely. Any isometry which is the identity at the boundary will give rise to the same measurements. Furthermore, the boundary distance function only takes into account the shortest paths, and it is easy to find counterexamples to unique determination, so one needs to pose some restrictions on the metric. Michel [Mi] conjectured that a simple metric g is uniquely determined, up to an action of a diffeomorphism fixing the boundary, by the boundary distance function pg(x,y) known for all x and y on dM. We recall Definition 1.1. We say that the Riemannian metric g is simple in M, if dM is strictly convex w.r.t. g, and for any x G M, the exponential map exp^ : exp~1(M) ? M is a diffeomorphism.