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On the characterization of the kernel of the geodesic X-ray transform

Author: Eduardo Chappa
Journal: Trans. Amer. Math. Soc. 358 (2006), 4793-4807
MSC (2000): Primary 58Jxx; Secondary 44A12, 53Cxx
Published electronically: June 20, 2006
MathSciNet review: 2231872
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Abstract: Let $ \overline{\Omega}$ be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space $ H^{k}(\overline{\Omega}), k \geq 1$. We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in $ H^{k}$.

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  • 1. Yu. E. Anikonov Some Methods for the Study of Multidimensional Inverse Problems Nauka, Sibirsk Otdel., Novosibirsk (1978).
  • 2. Yu. E. Anikonov and V. G. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl. 5 (1997), no. 6, 487–490 (1998). MR 1623603,
  • 3. I.N. Bernstein, M.L. Gerver Conditions of distinguishability of metrics by hodographs Methods and algorithms of interpretation of seismological information. Computerized Seismology. Vol. 13. Nauka, Moscow 50-73 (1980).
  • 4. J. Chazarain, A. Piriou Introduction to the Theory of Linear Partial Differential Equations. Studies in Mathematics and its Applications. Volume 14. North-Holland Publishing Company. MR 0678605 (83j:35001)
  • 5. Louis Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), no. 1-2, 11–51. MR 0407904,
  • 6. Peter B. Gilkey, The index theorem and the heat equation, Publish or Perish, Inc., Boston, Mass., 1974. Notes by Jon Sacks; Mathematics Lecture Series, No. 4. MR 0458504
  • 7. L. Hörmander The Analysis of Linear Partial Differential Operators Grund. der Math. Wiss. 256, Springer-Verlag (1983) MR 0717035 (85g:35002a)
  • 8. R.G. Mukhometov On the Problem of Integral Geometry Math. Problems in Geophysics. Akad. Nauk. SSSR, Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, 6, No. 2, 212-242 (1975).
  • 9. R.G. Mukhometov On a problem of reconstructing Riemannian metrics Sibirsk. Mat. Zh. 22, no. 3, 119-135, 237 (1981). MR 0621466 (82m:53071)
  • 10. L.N. Pestov, V.A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature Siberian Math. J. 29, no. 3, 114-130 (1988). MR 0953028 (89k:53066)
  • 11. V. A. Sharafutdinov, Integral geometry of a tensor field on a manifold with upper-bounded curvature, Sibirsk. Mat. Zh. 33 (1992), no. 3, 192–204, 221 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 3, 524–533 (1993). MR 1178471,
  • 12. V. A. Sharafutdinov, Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR 1374572
  • 13. Vladimir A. Sharafutdinov, Finiteness theorem for the ray transform on a Riemannian manifold, Inverse Problems 11 (1995), no. 5, 1039–1050. MR 1353801
  • 14. V.A. Sharafutdinov Ray Transform on Riemannian Manifolds University of Washington (1999).
  • 15. Vladimir Sharafutdinov and Gunther Uhlmann, On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points, J. Differential Geom. 56 (2000), no. 1, 93–110. MR 1863022
  • 16. P. Stefanov and G. Uhlmann Unpublished Notes, 1998.

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Additional Information

Eduardo Chappa
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Address at time of publication: Department of Mathematical and Physical Sciences, Texas A&M International University, Laredo, Texas 78041-1900

Received by editor(s): December 20, 2002
Received by editor(s) in revised form: August 3, 2004
Published electronically: June 20, 2006
Additional Notes: This work was partially supported by NSF grant #DMS-00-70488 and NSF grant #DMS-9705792
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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