Inventiones mathematicae

, Volume 193, Issue 1, pp 229–247

Tensor tomography on surfaces

  • Gabriel P. Paternain
  • Mikko Salo
  • Gunther Uhlmann
Article

Abstract

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

References

  1. 1.
    Anikonov, Yu., Romanov, V.: On uniqueness of determination of a form of first degree by its integrals along geodesics. J. Inverse Ill-Posed Probl. 5, 467–480 (1997) MathSciNetGoogle Scholar
  2. 2.
    Dairbekov, N.S.: Integral geometry problem for nontrapping manifolds. Inverse Probl. 22, 431–445 (2006) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dos Santos Ferreira, D., Kenig, C.E., Salo, M., Uhlmann, G.: Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178, 119–171 (2009) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the 1978 original MATHGoogle Scholar
  5. 5.
    Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved 2-manifolds. Topology 19, 301–312 (1980) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators. Classics in Mathematics. Springer, Berlin (2009). Reprint of the 1994 edition MATHCrossRefGoogle Scholar
  7. 7.
    Ivanov, S.: Volume comparison via boundary distances. In: Proceedings of the International Congress of Mathematicians, New Delhi, vol. II, pp. 769–784 (2010) Google Scholar
  8. 8.
    Michel, R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65, 71–83 (1981) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Mukhometov, R.G.: The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry. Dokl. Akad. Nauk SSSR 232(1), 32–35 (1977) (Russian) MathSciNetGoogle Scholar
  10. 10.
    Paternain, G.P.: Transparent connections over negatively curved surfaces. J. Mod. Dyn. 3, 311–333 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Paternain, G.P., Salo, M., Uhlmann, G.: The attenuated ray transform for connections and Higgs fields. Geom. Funct. Anal. 22, 1460–1489 (2012) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Pestov, L.: Well-Posedness Questions of the Ray Tomography Problems. Siberian Science Press, Novosibirsk (2003) (Russian) Google Scholar
  13. 13.
    Pestov, L., Sharafutdinov, V.A.: Integral geometry of tensor fields on a manifold of negative curvature. Sib. Math. J. 29, 427–441 (1988) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Pestov, L., Uhlmann, G.: On characterization of the range and inversion formulas for the geodesic X-ray transform. Int. Math. Res. Not. 4331–4347 (2004) Google Scholar
  15. 15.
    Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 161, 1089–1106 (2005) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Salo, M., Uhlmann, G.: The attenuated ray transform on simple surfaces. J. Differ. Geom. 88, 161–187 (2011) MathSciNetMATHGoogle Scholar
  17. 17.
    Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994) CrossRefGoogle Scholar
  18. 18.
    Sharafutdinov, V.A.: Integral geometry of a tensor field on a surface of revolution. Sib. Math. J. 38, 603–620 (1997) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sharafutdinov, V.A.: A problem in integral geometry in a nonconvex domain. Sib. Math. J. 43, 1159–1168 (2002) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sharafutdinov, V.A.: Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds. J. Geom. Anal. 17, 147–187 (2007) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Sharafutdinov, V.A., Skokan, M., Uhlmann, G.: Regularity of ghosts in tensor tomography. J. Geom. Anal. 15, 517–560 (2005) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Singer, I.M., Thorpe, J.A.: Lecture Notes on Elementary Topology and Geometry. Undergraduate Texts in Mathematics. Springer, New York (1976). Reprint of the 1967 edition MATHCrossRefGoogle Scholar
  23. 23.
    Stefanov, P., Uhlmann, G.: Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123, 445–467 (2004) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Stefanov, P., Uhlmann, G.: Boundary and lens rigidity, tensor tomography and analytic microlocal analysis. In: Aoki, T., Majima, H., Katei, Y., Tose, N. (eds.) Algebraic Analysis of Differential Equations, Festschrift in Honor of Takahiro Kawai, pp. 275–293 (2008) CrossRefGoogle Scholar
  25. 25.
    Stefanov, P., Uhlmann, G.: Linearizing non-linear inverse problems and its applications to inverse backscattering. J. Funct. Anal. 256, 2842–2866 (2009) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Thorbergsson, G.: Closed geodesics on non-compact Riemannian manifolds. Math. Z. 159, 249–258 (1978) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gabriel P. Paternain
    • 1
  • Mikko Salo
    • 2
  • Gunther Uhlmann
    • 3
    • 4
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA
  4. 4.Department of MathematicsUniversity of CaliforniaIrvineUSA

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