Boundary rigidity and stability for generic simple metrics

By Plamen Stefanov, Gunther Uhlmann. 2004

180 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40ff8a97ee7b508fc5d6e406d4a69a24bae033c1

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Citations in the corpus

Sensitivity analysis of an inverse problem for the wave equation with caustics (2012, result): “Moreover, our result is fundamentally different from those in the literature where Lipschitz, Hölder or logarithmic estimates are derived; see, for example, [6], [1], [33], [25], [26], [27], [28], [19] and [14].”
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions (2014, background): “It is also known that Im is always sinjective on simple surfaces [38], and I2 is s-injective for a generic class of simple metrics including real-analytic ones [52], but it remains an open question whether I2 is s-injective on arbitrary simple manifolds of dimension ≥ 3 (see however the recent paper”
Inverse problems: seeing the unseen (2014, background): “Finally, in [181] it is proven a conditional stability estimate of Hölder type.”
Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge (2017, background): “The most general result in this direction (outside a fixed conformal class, the setting of [29]) is the generic local (with respect to the metric) one proven in [25], i.”
Inverse scattering problems with multi-frequencies (2015, background): “Interestingly, the Lipschitz stability results were obtained in [30, 31, 93, 94] for the time-domain acoustic wave equation which is equivalent to availability of the data corresponding to all frequencies.”
Boundary rigidity and filling volume minimality of metrics close to a flat one (2010, background): “There are many more examples of locally rigid metrics: for instance, simple almost nonpositively curved metrics and simple analytic metrics are locally rigid [10], [19].”
Lens rigidity for manifolds with hyperbolic trapped set (2014, unknown): “This was proved by Stefanov-Uhlmann [StUh1].”
A Support Theorem for the Geodesic Ray Transform on Functions (2007, methodology): “Now by the work of Stefanov-Uhlmann in [15], we have that II is an elliptic analytic pseudodifferential operator.”
The geodesic X-ray transform with fold caustics (2010, unknown): “, [17, 3, 4, 24, 22, 26].”