Duke Mathematical Journal

Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Plamen Stefanov and Gunther Uhlmann

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the boundary rigidity problem for domains in Rn: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform Ig. We prove that the normal operator Ng=I*gIg is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

Article information

Duke Math. J. Volume 123, Number 3 (2004), 445-467.

First available in Project Euclid: 11 June 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
Secondary: 53C24: Rigidity results 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 35R30: Inverse problems


Stefanov, Plamen; Uhlmann, Gunther. Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123 (2004), no. 3, 445--467. doi:10.1215/S0012-7094-04-12332-2. http://projecteuclid.org/euclid.dmj/1086957713.

Export citation


  • Yu. Anikonov and V. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl. 5 (1997), 487–480.
  • I. N. Bernstein and M. L. Gerver, “Conditions on distinguishability of metrics by hodographs (in Russian)” in Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, 1980, 50–73.
  • E. Chappa, On the characterization of the kernel of the geodesic X-ray transform for tensor field, to appear in Trans. Amer. Math. Soc.
  • K. C. Creager, Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature 356 (1992), 309–314.
  • C. B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), 150–169.
  • –. –. –. –., Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), 445–464.
  • C. B. Croke, N. S. Dairbekov, and V. A. Sharafutdinov, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352 (2000), 3937–3956.
  • G. Eskin, Inverse scattering problem in anisotropic media, Comm. Math. Phys. 199 (1998), 471–491.
  • M. Gromov, Filling Riemannian manifolds, J. Differential Geometry 18 (1983), 1–147.
  • V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, 1977.
  • G. Herglotz, Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys. 52 (1905), 275–299.
  • M. Lassas, V. Sharafutdinov, and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann. 325 (2003), 767–793.
  • R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981), 71–83.
  • R. G. Muhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (in Russian), Dokl. Akad. Nauk SSSR 232, no. 1 (1977), 32–35.; English translation in Soviet Math. Dokl. 18, no. 1 (1977), 27–31.
  • –. –. –. –., On a problem of reconstructing Riemannian metrics, Siberian Math. J. 22, no. 3 (1981), 420–433.
  • R. G. Muhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an $n$-dimensional space (in Russian), Dokl. Akad. Nauk SSSR 243, no. 1 (1978), 41–44.
  • J.-P. Otal, Sur les longueur des géodésiques d'une métrique à coubure négative dans le disque, Comment. Math. Helv. 65 (1990), 334–347.
  • L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature (in Russian), Sibirsk. Mat. Zh. 29, no. 3 (1988), 114–130.; English translation in Siberian Math. J. 29, no. 3 (1988), 427–441.
  • V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser., VSP, Utrecht, Netherlands, 1994.
  • –. –. –. –., Finiteness theorem for the ray transform on a Riemannian manifold, Inverse Problems 11 (1995), 1039–1050.
  • V. Sharafutdinov and G. Uhlmann, On deformation boundary rigidity and spectral rigidity for Riemannian surfaces with no focal points, J. Differential Geom. 56 (2000), 93–110.
  • P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett. 5 (1998), 83–96.
  • M. E. Taylor, Pseudodifferential Operators, Princeton Math. Ser. 34, Princeton Univ. Press, Princeton, 1981.
  • E. Wiechert and K. Zoeppritz, Über Erdbebenwellen, Koenigl. Nachr. Gesellsch. Wiss. Goettingen 4 (1907), 415–549.