Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $H^{k}$.

References [Enhancements On Off] (What's this?)

  • Yu. E. Anikonov Some Methods for the Study of Multidimensional Inverse Problems Nauka, Sibirsk Otdel., Novosibirsk (1978).
  • 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, DOI
  • 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).
  • Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
  • Louis Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), no. 1-2, 11–51. MR 407904, DOI
  • 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
  • Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
  • 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).
  • R. G. Muhometov, On a problem of reconstructing Riemannian metrics, Sibirsk. Mat. Zh. 22 (1981), no. 3, 119–135, 237 (Russian). MR 621466
  • L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Sibirsk. Mat. Zh. 29 (1988), no. 3, 114–130, 221 (Russian); English transl., Siberian Math. J. 29 (1988), no. 3, 427–441 (1989). MR 953028, DOI
  • 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, DOI
  • V. A. Sharafutdinov, Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR 1374572
  • Vladimir A. Sharafutdinov, Finiteness theorem for the ray transform on a Riemannian manifold, Inverse Problems 11 (1995), no. 5, 1039–1050. MR 1353801
  • V.A. Sharafutdinov Ray Transform on Riemannian Manifolds University of Washington (1999).
  • 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
  • P. Stefanov and G. Uhlmann Unpublished Notes, 1998.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 58Jxx, 44A12, 53Cxx

Retrieve articles in all journals with MSC (2000): 58Jxx, 44A12, 53Cxx

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.