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

DOI:
https://doi.org/10.1090/S0002-9947-06-04059-1

Published electronically:
June 20, 2006

MathSciNet review:
2231872

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Abstract: Let be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space . 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 .

**1.****Yu. E. Anikonov***Some Methods for the Study of Multidimensional Inverse Problems*Nauka, Sibirsk Otdel., Novosibirsk (1978).**2.****Yu. E. Anikonov, V.G. Romanov***On uniqueness of determination of a form of first degree by its integrals along geodesics.*J. Inverse Ill-Posed Probl. 5, no. 6, 487-490 (1997). MR**1623603 (99f:53072)****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.****L. Boutet de Monvel***Boundary problems for pseudo-differential operators*Acta Math. 126, 11-51 (1971). MR**0407904 (53:11674)****6.****P. Gilkey***The Index Theorem and the Heat Equation*Princeton University (1974). MR**0458504 (56:16704)****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 whose curvature is bounded above*Siberian Math. J. 33, no 3, 192-204 (1992). MR**1178471 (94d:53116)****12.****V.A. Sharafutdinov***Integral Geometry of Tensor Fields*Inverse and Ill-Posed Problems Series VSP (1994). MR**1374572 (97h:53077)****13.****V.A. Sharafutdinov***Finiteness theorem for the ray transform on Riemannian manifolds*Inverse Problems 11, no. 5, 1039-1050 (1995). MR**1353801 (97a:58186)****14.****V.A. Sharafutdinov***Ray Transform on Riemannian Manifolds*University of Washington (1999).**15.****V. Sharafutdinov and G. Uhlmann***On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points*, J. Differential Geometry 56, 93-110 (2000). MR**1863022 (2002i:53056)****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

DOI:
https://doi.org/10.1090/S0002-9947-06-04059-1

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.