The Journal of Geometric Analysis

, Volume 15, Issue 3, pp 499–542 | Cite as

Regularity of ghosts in tensor tomography

  • Vladimir SharafutdinovEmail author
  • Michal Skokan
  • Gunther Uhlmann


We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L2-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.

Math Subject Classification

44A12 53A45 58J40 

Key Words and Phrases

Integral geometry ray transform pseudodifferential operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Chappa, E. On a characterization of the kernel of the geodesic X-ray transform,Trans. AMS, to appear.Google Scholar
  2. [2]
    Chazarain, J. and Piriou, A. Introduction to the theory of linear partial differential equations,Stud. Math. Appl. 14, North-Holland Publishing Company, (1982).Google Scholar
  3. [3]
    Dairbekov, N. Deformation boundary rigidity for SGM-manifolds and integral geometry problems for nontrapping manifolds, MPI-Preprint Nr. 04-06, Bonn, 14 p., (2004).Google Scholar
  4. [4]
    Graham, R., Knut, D., and Patashik, O.Concrete Mathematics, Addison-Wesley Publishing Co., Reading, MA, (1989).zbMATHGoogle Scholar
  5. [5]
    Pestov, L.Well-Posedness Questions of the Ray Tomography Problems, (Russian), Siberian Science Press, Novosibirsk, (2003).Google Scholar
  6. [6]
    Prudnikov, A., Brychkov, Y., and Marichev, O.Integrals and Series 1, Elementary Functions, New York a.o, London and Breach Publ., (1986).zbMATHGoogle Scholar
  7. [7]
    Sharafutdinov, V.Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, (1994).Google Scholar
  8. [8]
    Sharafutdinov, V. Finiteness theorem for the ray transform on a Riemannian manifold,Inverse Problems 11, 1039–1050, (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Sharafutdinov, V. Geometric symbol calculus for pseudodifferential operators, I,Siberian Adv. Math. 15(3), 81–125, (2005).MathSciNetGoogle Scholar
  10. [10]
    Sharafutdinov, V. and Uhlmann, G. On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points,J. Differential Geom. 56, 93–110, (2000).zbMATHMathSciNetGoogle Scholar
  11. [11]
    Stefanov, P. and Uhlmann, G. Stability estimates for the X-ray transform of tensor fields and boundary rigidity,Duke Math. J. 123, 445–467, (2004).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2005

Authors and Affiliations

  • Vladimir Sharafutdinov
    • 1
    Email author
  • Michal Skokan
    • 2
  • Gunther Uhlmann
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of MathematicsUniversity of WashingtonSeattle

Personalised recommendations